Rigid analytic curves and their Jacobians - OPARU

Transcription

Rigid analytic curves and their
Jacobians
Dissertation
zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät
für Mathematik und Wirtschaftswissenschaften der
Universität Ulm
vorgelegt von
Sophie Schmieg
aus
Ebersberg
Ulm 2013
Erstgutachter:
Prof. Dr. Werner Lütkebohmert
Zweitgutachter:
Prof. Dr. Stefan Wewers
Amtierender Dekan:
Prof. Dr. Dieter Rautenbach
Tag der Promotion: 19. Juni 2013
Contents
Glossary of Notations
vii
Introduction
1.
The Jacobian of a curve in the complex case . . . .
2.
Mumford curves and general rigid analytic curves .
3.
Outline of the chapters and the results of this work
4.
Acknowledgements . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
ix
ix
ix
x
xi
1. Some background on rigid geometry
1.1. Non-Archimedean analysis . . . . . . . . . . . . .
1.2. Affinoid varieties . . . . . . . . . . . . . . . . . .
1.3. Admissible coverings and rigid analytic varieties .
1.4. The reduction of a rigid analytic variety . . . . .
1.5. Adic topology and complete rings . . . . . . . . .
1.6. Formal schemes . . . . . . . . . . . . . . . . . . .
1.7. Analytification of an algebraic variety . . . . . .
1.8. Proper morphisms . . . . . . . . . . . . . . . . .
1.9. Étale morphisms . . . . . . . . . . . . . . . . . .
1.10. Meromorphic functions . . . . . . . . . . . . . . .
1.11. Examples . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
2
3
4
5
9
11
12
13
14
15
.
.
.
.
.
.
.
17
17
17
22
23
27
29
31
.
.
.
.
.
33
33
35
37
41
44
2. The
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
.
.
.
.
.
.
.
.
.
.
.
structure of a formal analytic curve
Basic definitions . . . . . . . . . . . . . . . . . . . .
The formal fiber of a point . . . . . . . . . . . . . .
The formal fiber of regular points and double points
The formal fiber of a general singular point . . . . .
Formal blow-ups . . . . . . . . . . . . . . . . . . . .
The stable reduction theorem . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . . . . .
3. Group objects and Jacobians
3.1. Some definitions from category theory
3.2. Group objects . . . . . . . . . . . . . .
3.3. Central extensions of group objects . .
3.4. Algebraic and formal analytic groups .
3.5. Extensions by tori . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4. The Jacobian of a formal analytic curve
51
4.1. The cohomology of graphs . . . . . . . . . . . . . . . . . . . . . . 51
4.2. The cohomology of curves with semi-stable reduction . . . . . . . 54
v
4.3. The Jacobian of a semi-stable curve . . . . . . . . . . . . . . . . 56
4.4. The Jacobian of a curve with semi-stable reduction . . . . . . . . 60
4.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A. Bibliography
vi
71
Glossary of Notations
We denote
R+
R+
0
Q(R)
R̊
Ř
K
R
k
Gm,K
Ḡm,K
Gm,k
BnK
X̃, f˜, x̃
by
the positive real numbers as a multiplicative group
the non-negative real numbers
the quotient field of a ring R
the subring {x ∈ R ; |x| ≤ 1} of a normed ring R
the ideal {x ∈ R ; |x| < 1} of R̊ where R is a normed ring
a valued field
the ring K̊
the field K̊/Ǩ
the multiplicative group of K, seen as an analytic variety
the group {x ∈ K ; |x| = 1} seen as a formal analytic variety
the multiplicative group of k, seen as an algebraic variety
the affinoid analytic variety Sp Khζ1 , . . . , ζn i
the reductions of the corresponding formal analytic counterpart.
vii
Introduction
In this work, we describe the general structure of a rigid analytic curve and its
Jacobian.
1. The Jacobian of a curve in the complex case
The study of the Jacobian of an algebraic curve started with the research of
certain integrals that appear in the calculation of the circumference of an ellipse.
Niels Henrik Abel and Carl Gustav Jacob Jacobi first described the Jacobian
variety around 1826. Of course, they could not formulate it in terms of algebraic
curves, since it was Bernhard Riemann, a good 25 years later, who first defined
the Riemann surface and thereby described algebraic curves over C. It took
many more men and women to arrive at the modern description of this theory
in the middle of the twentieth century.
The Jacobian variety of an algebraic curve of genus g is the set of line bundles
of degree zero over this curve. The tensor product gives this set the natural
structure of a group. A lot harder to show, but in the same way natural is its
structure as an algebraic variety of dimension g itself, containing the original
curve as a closed subvariety. The Jacobian variety is therefore an Abelian
variety with a canonical polarization, deriving from the embedding of the curve.
Over C the Jacobian variety of a curve of genus g can be described as
0
H (X, Ω1X/C )0 /H1 (X, Z), a quotient of Cg by a lattice M of rank 2g, which
is generated by certain integrals on the curve. We can write M = Zg ⊕ ZZg ,
so applying the exponential function let us write Ggm,C / exp(2πiZ). So the Jacobian of a Riemann surface is the multiplicative group of C to the power g
modulo a lattice of rank g.
2. Mumford curves and general rigid analytic curves
The complex numbers are just one possibility to create a topological and algebraic closure of Q. For every prime p we can define |x| = p−ν(x) , where ν is the
valuation associated to p. This was first described by Kurt Hensel and later
refined by his student Helmut Hasse at the end of the eighteenth century. The
topological closure of Q with this absolute value is the field Qp of p-adic numbers. Its algebraic closure Cp has infinite degree over Qp . There is no equivalent
of the exponential function on Cp and the topology has very strange properties.
Furthermore, the p-adic value gives rise to the reduction functor giving a close
relation to the finite field Fp and its algebraic closure.
The description of the Jacobian variety of a rigid analytic curve, i.e. a curve
over Qp mainly decomposes into two parts, a combinatorial one and a so-called
ix
formal one. Omitting the formal part, one can describe the Jacobian of Mumford curves, where the reduction has a certain simple form, as Ggm,K /M , where
M is a lattice of rank g, and thus showing a wonderful analogy to the complex case. Since the integral can not be defined over the p-adic numbers in a
meaningful way, one needs to replace the classic formulation of the Riemann
relations with a more general, cohomological one.
To describe the Jacobian of a general rigid analytic curve, one needs to work
with Raynaud extensions, heavily researched by Michel Raynaud, Siegfried
Bosch and Werner Lütkebohmert. Then one realizes that the Jacobian of a
rigid analytic curve can be written as E/M , where M is a lattice of rank g and
E is an extension
0 → Gtm,K → E → B → 0
of a formal analytic abelian variety B by the torus Gtm,K .
3. Outline of the chapters and the results of this work
In the first chapter, we will recapitulate the basic facts about rigid analytic
varieties, their topology and their relationship with formal analytic schemes
of locally topologically finite type. This chapter is by no means a complete
introduction into the topic, we refer to [BGR84] and [Bos05] for this.
In the second chapter, we will provide some new insight into the stable reduction theorem, by refining the proofs of a few theorems of [BL85], using much
shorter and less technical arguments than the original work. First, we will
show that the ring O̊X (X+ (x̃)) of bounded functions on the formal fiber of a
point x̃ of the reduction is local and henselian, by taking a close look at the
normalization of x̃.
Secondly we will then be able to give a much more natural proof for the
Theorem 2.4.1 which describes the periphery of a formal fiber. That periphery always consists of a disjoint union of annuli, which we can equate to the
structure of the normalization of the curve at the point x̃.
Finally, we will describe how this theorem is used to get to the stable reduction theorem.
In the third chapter, we will introduce group objects over an arbitrary category. This allows us to form a general theory describing algebraic, rigid analytic
and formal analytic groups simultaneously. With this theory, we can generalize
the results of [Ser88] and describe how an extension
0 → Gtm,K → E → B → 0
of an analytic or algebraic group B by the torus Gtm,K equals to a line bundle
over B.
The work of the third chapter pays off in the fourth and final chapter, where
we can give the explicit description of the lattice M which E gets divided by to
form the Jacobian variety of our curve. While it was known that such a lattice
exists and has full rank, we can even give a constructive formula for it. It will
be shown that the formal analytic variety B does not influence the absolute
x
value of the lattice and that one gains the explicit formula for the generators
vi = (vij )tj=1 of the lattice as
vij =
X
−dn · log|qn |
en ∈γi ∩γj
where qn is the height of the formal fiber of the double point corresponding to en
and dn = 1 if en has the same orientation in γi as in γj and dn = −1 otherwise
and where γi and γj are simple cycles on the curve, in close relationship to the
complex case.
The lattice can be described fully by this method, but the description depends
on the structure of the variety B of which little is known. We can construct B
explicitly for the special family of curves X, which have a reduction consisting
of a rational curve and curve Ỹ of genus g, together with a surjective map
ϕ : X → Y , with Y being a lift of Ỹ . Then it turns out that B is isogenic to
Jac Y of the same degree as the map ϕ.
4. Acknowledgements
This work would not have been possible without the help and support of many
people. I would like to thank my advisor Werner Lütkebohmert for his general
and huge insight in this topic and his willingness to share it with me.
I also want to thank Stefan Wewers for being the second expert for this thesis.
Many people have had part in teaching me mathematics. First and foremost
I want to thank my parents, who put their own studies at good use. Secondly,
I owe my thanks to my high school teacher Herbert Langer, who managed
to sneak some proofs and exactness into an otherwise quite boring and vague
curriculum and who organized extra maths classes for the interested. There
were many interesting courses at university, but I want to highlight the course
in analytic number theory by Helmut Maier and the courses in complex analysis
and advanced algebra by Werner Lütkebohmert. These two teachers greatly
influenced my understanding of mathematics and finally encouraged me to write
my dissertation in this field.
I also owe thanks to Irene Bouw and to all my colleagues of the department
for various help in teaching, research and life in general.
Further thanks go to the chairman of the examination board Werner Kratz
and his fellow examiners.
When writing a thesis in a foreign language, proofreading is even more important. So therefore I want to thank my former neighbor Angela Reichmeyer
for doing this – as she put it – almost meditative work.
Writing a thesis is a quite stressful task and good friends are hard to come
by. I am very glad that I can call Katja Setzer my friend, who always had an
open ear for various discussions.
Last but by no means least I like to express my sincere thanks to my parents
Cornelia and Johannes Schmieg. They brought me up, always supporting my
insatiable curiosity for the inner workings of everything.
xi
1.1. Non-Archimedean analysis
We need the notion of a non-Archimedean normed ring.
Definition 1.1.1. A ring A together with a function |·| : A → R+
0 is called a
normed ring if
1.
2.
3.
4.
|x| = 0 if and only if x = 0,
|1| = 1,
|xy| ≤ |x| · |y| for all x, y ∈ A,
|x − y| ≤ max(|x|, |y|) for all x, y ∈ A.
A normed ring is called valued if |xy| = |x| · |y| for all x, y ∈ A.
An A-module M together with a map k·k : M → R+
0 over a valued ring is
called a normed A-module if the following holds
1. kxk = 0 if and only if x = 0,
2. krxk = |r| · kxk for all r ∈ A, x ∈ M ,
3. kx + yk ≤ max(kxk, kyk) for all x, y ∈ M .
Note that valued rings are always integral and normed modules are always
torsion free. For any normed object T we denote by T̊ the set {x ∈ T ; |x| ≤ 1}
and by Ť the set {x ∈ T ; |x| < 1}. Note that T̊ is again a normed ring/module
and that Ť of a valued ring is a prime ideal in T̊ . This leads to the definition
of T̃ = T̊ /Ť .
Definition 1.1.2. A direct sum of normed modules M = M1 ⊕ M2 is called
orthonormal, denoted by M = M1 ⊥ M2 if
k(m1 , m2 )k = max(km1 k, km2 k)
If R is a valued ring, the quotient field Q(R) is also a valued ring by defining
a |a|
.
=
b
|b|
Proposition 1.1.3. Let A be a valued K-algebra over a valued field K with
|A| = |K|. Suppose M = M1 ⊕ M2 is a normed module over A, with M1 and
M2 being normed modules and we further have kM k = |A|. The direct sum is
orthonormal if and only if M̃ = M̃1 ⊕ M̃2 over the ring Ã.
1
Proof. It is clear that we have M̃ = M̃1 + M̃2 in any case, since the reduction
is surjective. For elements m̃1 ∈ M̃1 , m̃2 ∈ M̃2 the case m̃1 + m̃2 = 0 implies
km1 + m2 k < 1 for any lift m1 , m2 , but M = M1 ⊥ M2 . So this means we have
max(km1 k, km2 k) < 1, i.e. m̃1 = m̃2 = 0.
For the only if part of the proof look at an element m = m1 + m2 ∈ M with
m1 ∈ M1 and m2 ∈ M2 . We can assume without restriction that km1 k ≥ km2 k.
Suppose at first that km1 k = 1 and km2 k ≤ 1. This implies m̃1 6= 0. Thus we
get m̃ 6= 0 which means kmk = 1.
For the general case take any a ∈ K with |a| = km1 k. Then km1 /ak = 1 and
km2 /ak ≤ 1. So we have km/ak = 1 which implies kmk = |a|, thus making the
sum orthonormal.
1.2. Affinoid varieties
We have two basic approaches to formal analytic varieties which both have their
merits and flaws. In the next few sections we will explain how to construct rigid
analytic varieties and associate a reduction with them.
For this let K be a valued field, i.e. a valued ring that is a field. The associated
ring of elements with value less or equal then one, is denoted by R = K̊ and
the residue field R/Ř is denoted by k.
X
Definition 1.2.1. A power series
ak ζ1k1 ·· · ··ζnkn is called strictly convergent
k∈Nn
if lim|m|→∞ |am | = 0. The ring of the strictly convergent power series in n
variables is called the Tate algebra Tn := Khζ1 , . . . , ζn i. For each f ∈ Tn we
define the Gauss norm as |f | = max|am |.
Definition 1.2.2. The quotient A := Tn /a of Tn by some ideal a with the
reduction epimorphism α is called an affinoid K-algebra. An affinoid variety
Sp A is the pair (MaxSpec A, A). On A we have the residue norm
|α(f )|α = inf |f − a|
a∈a
and the supremum norm
|f |sup =
sup
|f (x)| ,
x∈MaxSpec A
where f (x) = f mod x for an maximal ideal x ∈ MaxSpec A as in algebraic
geometry.
It should be noted that there is always a finite field extension of K such that
there is an epimorphism α : Tn → A with |·|α = |·|sup .
Definition 1.2.3. Let A be an affinoid K-algebra and X := Sp A its affinoid
variety. For f1 , . . . , fr , g ∈ A without common zeroes the subset
X(|f1 /g|, . . . , |fr /g| ≤ 1) := {x ∈ X ; |fi (x)| ≤ |g|, i = 1, . . . , r}
is called a rational subdomain of X. If g = 1, it is called a Weierstrass domain.
If ε ∈ |K × | is a constant, we write X(|f | ≤ ε) or respectively X(|g| ≥ ε) for
the corresponding rational subdomain.
2
1.3. Admissible coverings and rigid analytic varieties
Definition 1.2.4. For an affinoid variety X := Sp A a subset U ⊂ X is called
an affinoid subdomain of X if there is an affinoid variety Y := Sp B and a
morphism ϕ : Y → X with ϕ(Y ) = U and for every morphism ϕ0 : Y 0 → X
with ϕ0 (X 0 ) ⊂ U there is a unique factorization ϕ0 = ψ ◦ ϕ.
A rational subdomain is an affinoid subdomain and according to Gerritzen’s
and Grauert’s theorem every affinoid subdomain is a finite union of rational
subdomains.
Proposition 1.2.5. Let X = Sp A be an affinoid variety. The restriction map
OX (X(|f | ≥ ε)) → OX (X(|f | = 1)) for an f ∈ O̊X (X) and any 0 < ε < 1 has
a dense image.
Proof. We know that OX (X(|f | ≥ ε)) = Ahεf −1 i and OX (X(|f | = 1)) =
Ahf −1 i. By construction, the ring A[f, f −1 ] is dense in the latter. But A[f, f −1 ]
is a subring of Ahεf −1 i as well, so the image of the restriction map is dense.
1.3. Admissible coverings and rigid analytic varieties
The topology induced by a Non-Archimedean absolute value has very bad properties. For example one can easily prove that such a topology is always completely disconnected. To counteract this behavior one needs to introduce the
concept of a Grothendieck topology.
Definition 1.3.1. Let X be a set and S ⊂ P(X) a subset of the power set of
X. Let further be {Cov U }U S
∈S be a family of coverings, i.e. a family of families
which satisfies Ui ⊂ U and i∈I Ui = U for every element {Ui }i∈I ∈ Cov(U ).
For a pair T = (S, {Cov U }U ∈S ) the elements of S are called admissible open
and the elements of Cov(U ) are called admissible coverings. T is called a Gtopology if it satisfies the following conditions.
1. U, V ∈ S ⇒ U ∩ V ∈ S.
2. U ∈ S ⇒ {U } ∈ Cov U .
3. If U ∈ S, {Ui }i∈I ∈ Cov U and {Vij }j∈Ji ∈ Cov Ui , then the covering
{Vij }i∈I,j∈Ji is also admissible.
4. If U, V ∈ S with U ⊂ V and {Vi }i∈I ∈ Cov V , then the covering {Vi ∩
U }i∈I of U is admissible.
The concept of a G-topology generalizes the usual definition of topologies.
Most topological concepts, like for example continuity, can directly be transferred to G-topologies by replacing open sets by admissible open ones. To
further add good properties one can define a unique finest slightly finer Gtopology of a given G-topology as in [BGR84, pg. 338 et seqq.]. In addition to
the stated axiom, the finest slightly finer topology will satisfy the following:
(G1) Any subset V of an admissible open set U for which an admissible covering
{Ui }i∈I of U with V ∩ Ui admissible open for every i ∈ I exists will be
admissible open.
3
(G2) A covering of an admissible open set consisting of admissible open sets
which has a refinement that is admissible is already admissible itself.
As a finer G-topology, the finest slightly finer G-topology will contain X and ∅
as admissible open sets if the original G-topology already did.
Definition 1.3.2. An affinoid variety X := Sp A carries the weak G-topology
T which is defined by declaring the affinoid subdomains admissible open and
defining admissible coverings as finite unions of affinoid subdomains.
The finest slightly finer G-topology of T is called the strong G-topology on
X. Both topologies contain ∅ and X as admissible open sets.
We can now define locally G-ringed spaces over a ring R as a set X with a
G-topology and a sheaf OX of algebras over R in the usual way. Note that an
affinoid variety with the strong topology is a locally G-ringed space over K.
Definition 1.3.3. A rigid analytic variety over K is a locally G-ringed space
(X, OX ) over K if it satisfies the following conditions.
1. The G-topology on X contains X and ∅ as admissible open sets and has
the properties (G1) and (G2).
2. There is an admissible covering {Ui }i∈I of X such that (Ui , OX |Ui ) is an
affinoid variety for all i ∈ I.
1.4. The reduction of a rigid analytic variety
We now want to generalize the concept of the reduction of a normed ring for
rigid analytic varieties.
Definition 1.4.1. Let X = Sp A be an affinoid variety. The affine scheme
X̃ = Spec Ã, with Ã = A/{f ∈ A ; |f | < 1} is called the reduction of X. The
map π : X → X̃ obtained by reducing maximal ideals is called the reduction
map.
An admissible open set U in X is called formal open if it is the preimage of
an Zariski open set under π.
Proposition 1.4.2. Let X = Sp A be an affinoid variety with irreducible reduction X̃ = Spec Ã and |A| = |K. Then the supremum norm |·| is multiplicative
on A.
Proof. Assume that there are elements f, g ∈ A such that |f · g| =
6 |f | · |g|.
We can set without restriction |f | = |g| = 1, since the absolute value is always
multiplicative for constants. So f, g ∈ Å and have non zero reductions f˜ and g̃.
But Ã is irreducible, so f˜ · g̃ 6= 0 which means that |f · g| = 1 in contradiction
with the hypothesis. So |·| is multiplicative.
Proposition 1.4.3. Let X = Sp A be an affinoid variety with reduction X̃ =
Spec Ã. Let Ỹ = Spec B̃ be an algebraic variety such that ϕ̃ : Ỹ → X̃ is a
smooth morphism and has finite presentation. There is an affinoid variety Y
together with a morphism ϕ : Y → X reducing to ϕ̃.
4
1.5. Adic topology and complete rings
Proof. By our assumption, we have B̃ = Ã[Z1 , . . . , Zn ]/(g̃1 , . . . , g̃r ). We take
any lift of g1 , . . . , gr in Ahζ1 , . . . , ζn i and set Y = Sp Ahζ1 , . . . , ζn i/(g1 , . . . , gr ).
Definition 1.4.4. An admissible open affinoid covering {Ui }i∈I of a rigid analytic variety X is called a formal covering if Ui ∩ Uj is formal open in Ui for
every i, j ∈ I.
If we are given a formal covering {Ui }i∈I of a rigid analytic variety X we can
define the reduction X̃ of X by reducing the Ui and gluing along the reduction of
Ui ∩ Uj . This reduction is not unique and depends on the used formal covering.
In chapter 2 we will construct coverings such that the reduction has very simple
singularities.
Proposition 1.4.5. Let X be a rigid analytic curve and U a formal covering
leading to the reduction π : X → X̃. Let Ũ0 = (Ũi0 )i∈I be an affine covering
of X̃. Then U0 = (π −1 (Ũi0 )i∈I is again a formal covering and the associated
reduction is X̃.
Proof. First, we need to see that U0 is admissible. Take an admissible open set
Ui ∈ U. Then Ũi ∩ Ũj0 is affine for every Uj0 ∈ U0 . Therefore there is a finite
set of elements fi ∈ OX (Ui ) such that Ui ∩ Uj0 = Ui (|fi | ≥ 1) so Ui ∩ Uj0 is a
rational subdomain of Ui and as such admissible. But then Uj0 is admissible by
(G1). Using this construction we also see that Ui0 is affinoid with reduction Ũi0 .
The covering U0 is also formal since Ui0 ∩ Uj0 is formal by the very definition of
U0 . Since the reduction of Ui ∩ U0 glues together to form Ũi we get that the
reduction associated to U0 is again X̃.
We now come to the second approach to formal analytic varieties.
Definition 1.5.1. Let R be a ring and a ⊂ R an ideal. The topology for which
ak is a basis of open neighborhoods of 0 is called the a-adic topology of R.
For a module M over R we define the a-adic topology of M to be the topology
with ak M as basis for 0.
The ideal a is called the ideal of definition.
The adic topology on a ring is very similar to a norm as the following proposition illustrates.
Proposition 1.5.2. An a-adic noetherian topological ring is normed with Ř =
a.
Proof. Assume that R is a-adic. For x ∈ R we define
(
0
if x = 0
|x| =
exp(−n) where n is the smallest integer with x 6∈ an+1
5
T
According to Krull’s intersection theorem [Bos05, 2.1.2], we know that an =
{0}. So this norm is well-defined and |x| = 0 implies that x = 0.
If x ∈ an and y ∈ am we clearly have xy ∈ an+m and x + y ∈ amin(n,m) so
|xy| ≤ |x| · |y| and |x + y| ≤ max(|x|, |y|). Furthermore we get a = Ř and for n
and ε with exp(−n) < ε ≤ exp(−n + 1) we have |r| < ε if and only if r ∈ an .
So both induce the same topology.
The converse of this proposition is not true, as one can easily see at the
example of a normed ring where Ř is generated by elements of different norm.
We can, however, show that valued valuation rings are always adic, as we will
do in the next few propositions.
Proposition 1.5.3. Let R be a ring with a-adic topology. Then
∞
X
xk
k=1
is a Cauchy sequence if and only if (xk ) is a sequence tending to zero.
P
if for every n ∈ N we have N ∈ N such
Proof. The series ∞
k=1 xk is Cauchy
P
that for every i, j > N the sum jk=i xk is in an . The case i = j implies that
(xk ) is a zero sequence as always.
n
If (xk ) tends
Pj to zero wenfind an N for every n such that xk ∈ a for k > N .
But then k=i xk is in a for every i, j > N as a finite sum of elements in
an .
Definition 1.5.4. An integral ring R in which for every element x ∈ Q(R)×
either x ∈ R or x−1 ∈ R holds is called a valuation ring.
Proposition 1.5.5. Let R be an integral ring. Then the following two statements are equivalent:
(i) R is a valued ring with R = Q̊(R) = {a/b ∈ Q(R) ; |a| ≤ |b|}.
(ii) R is a valuation ring with Krull dimension 1.
Proof. Suppose R is a valued ring with R = Q̊(R). We can extend the absolute
value of R to an absolute value of Q(R) by defining
a
|a|
.
:=
b
|b|
Then if x ∈ Q(R) we either have |x| ≤ 1 which implies x ∈ R by our assumption
or |x−1 | = |x|−1 < 1 which induces x−1 ∈ R.
If p is a prime ideal not equal to zero we have x ∈ p with x 6= 0. For any
y ∈ Ř we can find an n ∈ N such that |y|n < |x|. Then y n /x ∈ R and we get
y n ∈ p which implies y ∈ p since p is prime. Since R× = {x ∈ R ; |x| = 1}
by the assumption that R = Q̊(R) this implies that p = Ř and R has Krull
dimension 1.
A valuation ring is always local according to [Bos05, 2.1.6]. Let t ∈ R \ {0}
be an arbitrary non-unit. The set b = {x ∈ R ; t−n x ∈ R for all n ∈ Z} is
6
a prime ideal in R, since xy ∈ b, x 6∈ b implies that there is an m ∈ Z with
t−m x 6∈ R and therefore tm x−1 ∈ R which implies t−n y = t−n−m tm x−1 xy ∈ R
for all n ∈ Z. Since t 6∈ b we get b = 0 by our assumption of Krull dimension
1. This means that
km,x = max(n ∈ Z ; t−n xm ∈ R)
is well-defined for all x ∈ R \ {0} and m ≥ 1. Furthermore we get km,x ≥ 0.
One can see directly that the limit of km,x /m exists and for every n ∈ N the
inequality
kn,x
km,x
kn,x + 1
≤ lim
≤
m→∞ m
n
n
holds. We set
km,x
|x| = lim exp −
.
m→∞
m
For x ∈ R× we can calculate km,x = 0 for all m ∈ N so |x| = 1. If x is not a unit,
then by the same argument as before there is an m ∈ N such that t−1 xm ∈ R
and therefore |x| 6= 1. We can define km,x for every x ∈ Q(R) and one easily
realizes that |xy −1 | = |x| · |y|−1 . So |·| : Q(R)× → R+ is a group morphism with
kernel R× and x ∈ R if and only if |x| ≤ 1.
Any valuation ring defines a valuation with the axioms of Definition 1.1.1 if
one replaces R+ with Q(R)× /R× imbued with the canonical ordering α ≤ β ⇔
αβ −1 ∈ R for α, β ∈ Q(R)× /R× . So |·| : Q(R)× → R+ makes R a valued ring
with R = Q̊(R).
Proposition 1.5.6. Let R be a valued ring with R = Q̊(R). Then the topology
of R is a-adic with a = (t) for any t ∈ R with |t| < 1.
Proof. The sets Bε := {x ∈ R ; |x| < ε} form a basis of the open neighborhoods
of 0 by the definition of the topology of a metric space. We need to show that
Bε is open in the a-adic topology. Since |t| < 1 we find n ∈ N with |t|n < ε.
Then for every x ∈ Bε and every r ∈ R we get |x + rtn | < ε and x + an is an
a-adic neighborhood of x contained in Bε .
On the other hand, let x = rtn ∈ an be any element. Choose an ε < |t|n .
Then |y − x| < ε implies |y/tn − r| < 1 and therefore |y/tn | ≤ 1. This means
that there is an r0 ∈ R such that y = r0 tn ∈ an and an is open.
Remark. It should be noted that even so a valued ring R with R = Q̊(R) is local,
the ideal generating its adic topology is not the maximal ideal of R. Indeed if
|Q(R)× | is dense in R+ we have Řn = Ř for any n ∈ N.
We can define a topology on every valuation ring that is not a field by using
all non-zero ideals as base for the topology. This topology is Hausdorffsch since
0 and t are separated by any ideal generated by an element smaller than t in
Q(R)× /R× . Such an element exists as t2 is smaller than t if t is not a unit and
any non-unit is smaller than any unit t. If there is an ideal a in R so that this
topology of R is a-adic we call R adic.
7
Proposition 1.5.7. Let R be an adic valuation ring with a finitely generated
ideal of definition. Then the ideal of definition is principal and there is a nontrivial minimal prime ideal p and the t-adic topology on R coincides with the
a-adic one for every t ∈ p.
Proof. We have already given the proof for the case most interesting to us,
namely when R is a valued ring. For arbitrary valuation ring the proof can be
found in [Bos05, 2.1.7].
Proposition 1.5.8. Let R be a Hausdorffsch adic ring with ideal of definition
a. The topological ring R is complete if and only if
R = lim R/an
←−
holds where R/an has the discrete topology.
Proof. Recall that we have
lim R/an = {(xn ) ∈
←−
∞
Y
R/an ; xn ≡ xm
mod am
for m ≤ n} .
n=1
By the definition of the inverse limit there is always a map ϕ : R → lim R/an .
←−
For noetherian rings this map is injective because of Krull’s intersection theorem, otherwise this is just the assumption that R is Hausdorffsch, so ϕ is
injective anyway.
If R is complete and x ∈ lim R/an is represented by a sequence (xn ) ∈
Q∞
←−
n
n mod an is a Cauchy
n=1 R/a then any sequence (xn ) ⊂ R with xn = x
sequence in R and its limit is x.
If (xk ) is a Cauchy sequence in lim R/an then for every n there is an N such
←−
that for every k, l ≥ N we have xk − xl = 0 mod an . We set xn = xN + an ∈
R/an . Then (xn ) represents an element x in lim R/an and (xk ) → x, so lim R/an
←−
←−
is complete.
Proposition 1.5.9. Let R be an adic valuation ring with ideal of definition a.
The ring R̂ = lim R/an is adic if a is finitely generated.
←−
Proof. As said before there is an injective map ϕ : R → R̂. We want to show
that R̂ is adic with ideal of definition aR̂. The topology on R̂ is the coarsest
topology on lim R/an such that every projection pn : R̂ → R/an is continuous.
←−
This means that (ker pn )n∈N is a base of neighborhoods
for 0.
Q
n
n
Let x ∈ ker pm be represented by (x ) ∈ R/a . Then xn = 0 mod am for
every n ≥ m. If we take any lift xn ∈ R of xn we have xn ∈ am and ϕ(xn )
converges to x. We have seen in Proposition 1.5.7 that we can assume a to be
principal, i.e. a = (t). This means we can write xn = rn tm for some rn ∈ R. But
xk ≡ xn mod ak for every n > k so rn ≡ rk mod ak−m for every n > k ≥ m.
m
n
This means (rk )∞
k=m represents some r ∈ R̂ and x = rϕ(t ). So ϕ(a )R̂ forms
a basis of the neighborhoods of 0 and R̂ is adic with ideal of definition aR̂.
Definition 1.5.10. We call R̂ the completion of R.
8
1.6. Formal schemes
1.6. Formal schemes
With these definitions in mind, we can now explain formal schemes.
Definition 1.6.1. Let A be a complete Hausdorffsch adic ring with principal
ideal of definition a. For a free variable ζ we call Ahζi = lim A/an [ζ] the ring
←−
of convergent power series in ζ over A.
Proposition 1.6.2. Let A be a complete adic ring with principal ideal of definition a. The elements of the ring Ahζi can uniquely be written as
∞
X
ak ζ k
k=0
with a zero sequence (ak ) ⊂ A.
Proof. The ring A[ζ] is adic with ideal of definition aA[ζ] by the definition of
the product topology. Therefore Ahζi isPthe completion of A[ζ]. If ak is a zero
k
sequence then so is ak ζ k and therefore ∞
k=0 ak ζ is in Ahζi.
If f ∈ Ahζi is any element, then there is a Cauchy sequence in A[ζ] converging
to f . This sequence gives a Cauchy sequence in A for every coefficient. But A
is complete so we can assume without restriction that the sequence
k-th
P∞for the
k
coefficient is constant for n > Nk . Therefore we can write f = k=0 ak ζ .
This proposition shows that the definition of Khζi given in Section 1.2 is
equal to this one.
Definition 1.6.3. Let A be a complete and Hausdorffsch adic ring with principal ideal of definition a. For f ∈ A we call Ahf −1 i = lim A/a[f −1 ] the complete
←−
localization of A by f .
Proposition 1.6.4. Let A be a complete and Hausdorffsch adic ring with principal ideal of definition a. Then Ahf −1 i is the adic completion of A[f −1 ] with
respect to the ideal aA[f −1 ]. There is a canonical isomorphism
Rhζi/(1 − f ζ) →
˜ Rhf −1 i
Proof. See [Bos05, Remark 2.1.8 and 2.1.9].
Definition 1.6.5. Let A be a complete and Hausdorffsch adic ring with principal ideal of definition a. The affine formal scheme Spf A is the space of all
open prime ideals in A with Spf Ahf −1 i as base of open sets together with
the structure sheaf A where A(Spf Ahf −1 i) = Ahf −1 i is obtained by complete
localization.
Remark. A prime ideal p in A is open if and only if an ⊂ p for some n ∈ N. This
implies that p is open exactly if a ⊂ p and we get a one-to-one correspondence
of open prime ideals in A and prime ideals in A/a.
One can show (see [Bos05, 2.2]) that Spf(A) is indeed a locally topologically
ringed space, i.e. that A is a structure sheaf.
9
Definition 1.6.6. A formal scheme is a locally topologically ringed space
(X, OX ) such that each point x ∈ X has an open neighborhood U such that
(U, OX (U )) is an affine formal scheme.
As per usual, one can construct a formal scheme by gluing affine formal
schemes. With the completed tensor product, one can define a fiber product in
the category of formal schemes. See, as usual [Bos05, Chapter 2] and [BGR84,
Part C] for further details.
An important case of formal schemes arise as the completion of algebraic
schemes. If X is any separated scheme and Y is a closed subscheme with ideal
of definition J in OX , then the completion of OX alongside J together with
the topological space Y gives a formal scheme.
Definition 1.6.7. Let R be an adic valuation ring with principal ideal of definition a. A topological R-algebra A is called of topologically finite type if A is
isomorphic to Rhζ1 , . . . , ζn i/b endowed with the topology induced by a.
A is called admissible, if furthermore b is finitely generated and an f = 0 for
any n ∈ N implies f = 0.
A formal scheme X is called admissible if there is an affine covering (Ui ) of
X such that Ui = Spf Ai and Ai is admissible.
If X is an integral, projective, flat R scheme, where R is a valuation ring,
then the completion of X along its special fiber gives an admissible formal R
scheme X. The scheme X is then called the analytification of X.
We want to study how admissible formal schemes are connected with rigid
analytic varieties with a fixed formal covering. For this, we introduce two
covariant functors.
Definition 1.6.8. There is a functor
rig : (admissible formal schemes over R) → (rigid analytic varieties over K)
defined by rig(Spf A) = Sp A ⊗R K on the affine formal schemes, associating a
formal scheme with its generic fiber, where K = Q(R).
Furthermore we have the reduction functor
red : (admissible formal schemes over R) → (algebraic varieties over k)
defined via red(Spf A) = Spec A/a where a is the ideal of definition of R and k
is the residue field of R.
To see that these functors are well-defined we again refer to [Bos05, 2.4.]. It
is easy to see that red(X) is indeed a reduction of rig(X) as this is clearly the
case if X is affine and is in general is obtained by using an admissible covering
of X. If X is an analytification of some integral, projective, flat R scheme X,
then rig(X) is the analytification of the generic fiber of X as we will define it
in Section 1.7.
On the other hand, if we have an affinoid variety XK = Sp Khζ1 , . . . , ζn i/bK ,
where b ⊂ Rhζ1 , . . . , ζn i is a finitely generated ideal and bK = b ⊗R K, then
X = Spf Rhζ1 , . . . , ζn i/b is a formal scheme which is automatically admissible
10
1.7. Analytification of an algebraic variety
by Noether normalization. The functor rig(X) obviously yields XK and red(X)
gives the canonical reduction X̃ of X. If X is a rigid analytic variety with formal
covering (Ui )i∈I , then the covering gives gluing relations for a formal scheme,
using the fact that Ũi ∩ Ũj is an open subset of Ũi .
These relations allow us to use admissible formal schemes and rigid analytic
varieties with fixed reduction interchangeably.
1.7. Analytification of an algebraic variety
One of the most important aspects of rigid analytic varieties is the possibility to
view any algebraic variety X over a non-Archimedean field K as a rigid analytic
variety over K.
We only sketch the process for affine and projective varieties. See [BGR84,
9.3.4] for a more detailed discussion.
In order to construct a rigid analytic variety X an out of the affine algebraic
variety X = Spec K[ξ1 , . . . , ξn ]/a we take any c ∈ K with |c| > 1 and set
Tn,k = Khc−k ξ1 , . . . , c−k ξn i .
The ring K[ξ1 , . . . , ξn ] is part of all the Tn,k , so we get a sequence of K-algebra
morphisms
Tn,0 /aTn,0 ← Tn,1 /aTn,1 ← . . . ,
which gives rise to a sequence of open immersions
Sp Tn,0 /aTn,0 → Sp Tn,1 /aTn,1 → . . . .
By pasting along these maps we can construct an analytic variety X an with
Sp Tn,k /aTn,k as admissible affinoid covering. However this covering is never
formal, since Sp Tn,k /aTn,k yields merely a finite set of points in the reduction
of Sp Tn,k+1 /aTn,k+1 .
In this work the most prominent example of an analytic variety constructed
this way is the variety Gm,K .
Proposition 1.7.1. The variety Gm,K which is the analytification of the affine
variety Spec K[ζ, ζ −1 ] is a rigid analytic group in the sense of Chapter 3. The
affinoid variety Ḡm,K = Sp Khζ, ζ −1 i is an open subgroup variety of Gm,K ,
consisting of the elements with absolute value 1 in K. Its reduction is Gm,k =
Spec k[Z, Z −1 ] with a group structure induced by Ḡm,K .
Proof. Since Spec K[ζ, ζ −1 ] is the algebraic description of Gm,K and morphisms
carry over in the analytification process we see that Gm,K is the analytic group
variety with the group structure of K × . The variety Ḡm,K is the first variety
of the defining sequence of Gm,K and as such Ḡm,K is an open subgroup of
Gm,K .
If X is projective, say X ⊂ Pn , then there are (n + 1) affine
Ui = Spec K[ζ0 /ζi , . . . , 1, . . . , ζn /ζi ]
11
in Pn for which we can find analytic counterparts. The gluing of the Ui is done
by equating ζk /ζi with ζk /ζj ·ζj /ζi where ζi /ζj and ζj /ζi are not null. But these
gluing relations are already defined for the affinoid varieties
Sp Khζ0 /ζi , . . . , 1, . . . , ζn /ζi i
where ever |ζi /ζj | = 1, so the analytification of Pn and the subset X can be
obtained by gluing n + 1 copies of BnK at their borders.
Since BnK is affinoid with reduction Ank and we glued at
Sp Khζ0 /ζi , . . . , 1, . . . , ζ0 /ζi , ζi /ζj i
which reduces to the correct gluing for Pnk , this also gives a formal covering of
X.
1.8. Proper morphisms
We want to give the definition of a proper analytic variety and show that the
analytification of a projective algebraic variety is proper. To be able to do so, we
need to formulate the concepts of separate morphisms and relatively compact
subsets.
Definition 1.8.1. A morphism ϕ : X → Y of analytic varieties is called separated, if the diagonal morphism ∆ : X → X ×Y X is a closed immersion.
One easily realizes as in [BGR84, 9.6] that morphisms of affinoid varieties
and therefore affinoid varieties over Sp K are always separated. Since the definition coincides with the algebraic one, it is easy to see that analytifications of
separated algebraic morphisms are again separated.
Definition 1.8.2. Let X = Sp A and Y = Sp B be affinoid varieties with a
morphism ϕ : X → Y . An affinoid subset U ⊂ X is said to be relatively compact
in X over Y if there exists an affinoid generating system f1 , . . . , fr of A over B
such that U ⊂ X(|f1 | < 1, . . . , |fr | < 1).
The relative p
compactness of a subset U ⊂ X is equivalent of assuming that
there is an ε ∈ |K × | with ε < 1 such that U ⊂ X(ε−1 f1 , . . . , ε−1 fr ).
Now we can define proper morphisms.
Definition 1.8.3. A morphism ϕ : X → Y of analytic varieties is called proper
if ϕ is separated and if there is an admissible affinoid covering (Yi )i∈I of Y such
that, for every i ∈ I there are two finite admissible affinoid coverings Xij and
Xij0 of ϕ−1 (Yi ) such that Xij is relatively compact in Xij0 over Yi for all indices
i and j.
Proposition 1.8.4. The analytification of a projective variety is proper.
Proof. The closed ball BnK is relatively compact in every ball of greater radius. Since a projective variety allows finite admissible coverings coming from
BnK with any radius greater or equal to one we get the necessary covering for
properness.
12
1.9. Étale morphisms
1.9. Étale morphisms
Definition 1.9.1. Let ϕ : X → Y be a morphism, x ∈ X a point with an
admissible open neighborhood U and an immersion ι : U → BnY . The morphism
ϕ is called smooth of relative dimension r at x if there are n − r sections
g1 , . . . , gn−r locally at y = ϕ(x) generating the ideal defining U as a subscheme
of BnY with dg1 , . . . , dgn−r being linearly independent in Ω1Bn /Y .
Y
Furthermore ϕ is called formal smooth of relative dimension r if the differential forms dg1 , . . . , dgn−r form an orthonormal system in Ω1Bn /Y .
Y
Definition 1.9.2. A morphism is called étale or formal étale if it is smooth or
formal smooth of relative dimension 0 respectively.
Definition 1.9.3. Let R be a local ring with residue field k. The ring R is
called henselian if any monic polynomial p ∈ R[T ] which admits a factorization
p̃ = f˜ · g̃ ∈ k[T ] with coprime factors f˜ and g̃ in the reduction has lifts of f and
g in R[T ] with p = f · g.
The smallest local ring extension Rh of R such that Rh is henselian is called
the henselization of R.
We mainly need the characterization of the henselization by étale morphisms
This is done by the next proposition.
Proposition 1.9.4. Let R be a local ring. The henselization Rh of R is the
direct limit limi∈I Ri of all isomorphism classes of R-algebras Ri which occur
−→
as local rings of some étale R-scheme at the closed point lying over the closed
point of R.
See [BLR90, 2.3]
The last proposition allows us to assume that the local ring OX,x is henselian
if one allows for étale base change.
Proposition 1.9.5. Let X be an algebraic curve over an algebraically closed
field k and let x ∈ X a closed point. There is an étale morphism X̂ → X such
that x lies on n different irreducible components of X̂, where n is the number
of points in the fiber of x in the normalization.
Proof. If x lies on n different irreducible components, then the normalization
of X has n disjoint components with points lying over x. Since according to
[Ray70, p. 99] henselization commutes with normalization we can assume that
x lies on exactly one irreducible component and show that there is only one
point in the normalization of X̂ over x.
Since we look at X up to étale morphism we can assume the local ring OX,x
to be henselian. Let y1 , . . . , yn be the points lying over x in its normalization
X 0 . There are gi ∈ OX 0 ,y1 ,...,yn such that p = (T − g1 ) · · · · · (T − gn ) ∈ OX,x [T ] is
a monic polynomial and that gi (yj ) = 1 − δij . Then p(x) = T · h where h ∈ k[T ]
is not divisible by T since we can calculate p(x) as p(yi ) for any i between 1
and n. But OX,x is henselian and any non trivial factorization of p is contrary
to our assumptions, which implies n = 1.
13
The lift of a morphism ϕ̃ as constructed in Proposition 1.4.3 is étale if ϕ̃ is
étale.
Proposition 1.9.6. Let X and Y be affinoid varieties with associated reductions πX : X → X̃ and πY : Y → Ỹ . Let Y be formal smooth over K. For every
morphism ϕ̃ : X̃ → Ỹ there is a lift ϕ : X → Y of ϕ̃.
Proof. We set X = Spf A and Y = Spf B, A = lim A ⊗ Ri , B = lim B ⊗
←−
←−
Ri , R = lim Ri with Ri = R/ai as before. By [BLR90, Prop. 2.2.6] we get
←−
HomR (Xi , Y ) HomR (Xi−1 , Y ) since Y → Spf R is smooth. Therefore the
map ϕ̃, which gives a map ϕ1 : X1 → Y successively lifts to maps ϕi : Xi → Y
and therefore ϕ : X → Y exists and has the proposed reduction.
Proposition 1.9.7. Let X and Y be affinoid varieties with reductions X̃ and
Ỹ . Let ϕ : X → Y be a formal smooth morphism admitting a section σ̃ : Ỹ → X̃
in the reduction. There is a lift of σ̃ which is a section of ϕ.
Proof. Again by [BLR90, Prop. 2.2.6] we get HomY (Yi , X) HomY (Yi−1 , X),
which lifts σ̃ to σ ∈ HomY (Y, X). By definition of HomY , σ is a section of
ϕ.
1.10. Meromorphic functions
Definition 1.10.1. Let X be a reduced rigid analytic variety and U an open
affinoid subvariety. The field of fractions Q(OX (U )) is called the field of meromorphic function over U . It extends to a sheaf MX of meromorphic functions
on X. The global sections of this sheaf are called meromorphic functions.
Proposition 1.10.2. Let X be a projective rigid analytic curve over an algebraically closed field K and U be an affinoid subvariety with irreducible reduction. Then the valuation of OX (U ) extends canonically to MX (U ).
Proof. Let f ∈ MX (U ) be a meromorphic function. There are functions
g, h ∈ OX (U ) with f = g/h by definition. We set |f | = |g|/|h|. According
to Proposition 1.4.2, the absolute value on U is multiplicative and as such well
defined.
We can define a reduction of a meromorphic function f if |f | = 1 on Xi =
π −1 (X̃i \ Sing X̃) by setting f˜ = g̃/h̃ which is a rational function on X̃i .
Proposition 1.10.3. Let X be a projective smooth rigid analytic curve with
reduction X̃. Let f˜ ∈ k(X̃) be a rational function. Then f˜ has a meromorphic
lift f ∈ MX (X).
Proof. Let f˜ be defined on Ũ ⊂ X̃. Let U := π −1 (U ), then we find a lift
f ∈ O̊X (U ) of f˜. The algebraic functions are dense in O̊X (U ) by definition
which lets us approximate f with algebraic functions on U and therefore algebraic rational functions on X. The limit of these functions therefore gives a
meromorphic lift of f .
14
1.11. Examples
Proposition 1.10.4. Let X be a projective rigid analytic curve and let f ∈
MX (X) be a non-constant meromorphic function. The set {x ∈ X ; |f (x)| ≤
1} is an affinoid variety.
Proof. The meromorphic function f : X → P1 defines a finite covering map of
X. Since B1 = {x ∈ P1 ; |x| ≤ 1} and {x ∈ P1 ; |x| ≥ 1} is an admissible
affinoid covering of P1 , the preimages of these sets are affinoid too.
1.11. Examples
Example 1.11.1. The unit ball BnK = Sp Khζ1 , . . . , ζn i = {x ∈ K n ; |xi | ≤ 1}
is an example for an affinoid variety. The unit ball reduces to the affine space
Ank .
Example 1.11.2. The affine space AnK can be seen as a rigid analytic variety
by covering it with balls with increasing radii. Note that any ball of strictly
smaller radius in a bigger ball reduces to a point, so this covering is not formal.
Example 1.11.3. The projective space PnK can be obtained by the usual way
of covering it with n + 1 copies of AnK . One sees that covering it with n + 1
copies of the unit ball which are identified by their border {x ∈ K ; |x| = 1} is
already enough. This covering is formal and the reduction obtained this way is
the Pnk .
A projective algebraic variety given by homogeneous equations in some PnK
therefore also admits a formal covering and its reduction according to this
covering can be computed by reducing the equations after normalization. We
denote such a rigid analytic variety as a projective analytic variety.
Example 1.11.4. Let q ∈ Gm,K with |q| < 1. The elliptic curve with parameter
q is defined as the quotient X = Gm,K /(q Z ). We want to construct an admissible
affinoid covering of X. Let
U1 := {x ∈ Gm,K ; q ≤ |x| ≤
√
q},
U2 := {x ∈ Gm,K ;
√
q ≤ |x| ≤ 1} ,
√
two annuli of height q. We can map Ui onto X by taking the points modulo
q. Both maps are analytic isomorphisms. Thus U = {U1 , U2 } is an admissible
affinoid cover of X. Since
√
U1 = Sp Kh q −1 ζ, qζ −1 i
and
√
U2 = Sp Khζ, qζ −1 i
we get
Ũi = Spec k[Xi , Yi ]/(Xi · Yi )
√
√
mapping q −1 ζ to X1 , qζ −1 to Y1 , ζ to X2 and qζ −1 to Y2 . The relation
q ≡ 1 gives us the gluing isomorphisms X1 = 1/Y2 and X2 = 1/Y1 . Thus X̃
has two components of genus 0, meeting in two double points.
15
√
The reduction map π maps points with absolute value between q and q
√
to the double point of Ũ1 , points with absolute value between q and 1 to
the double point of Ũ2 . Points with absolute value 1 are mapped onto the
√
component X1 = Y2 = 0 and those with absolute value q are mapped onto
the component X2 = Y1 = 0. In Chapter 4 we will show that this curve is
indeed an elliptic curve.
16
2. The structure of a formal analytic
curve
In this chapter, we will analyze the structure of a p-adic curve and its dependence on the structure of its reduction.
2.1. Basic definitions
Definition 2.1.1. Let K be an algebraically and topologically complete nonArchimedean valued field. An analytic variety X over K of dimension 1 with a
fixed reduction π : X → X̃ is called a formal analytic curve if X is smooth over
K. Therefore OX (U ) is a reduced and irreducible K algebra of dimension 1.
The first important to mention fact about analytic curves is that according
to [BGR84, 6.4.3/1], over an algebraically closed field K, we can assume that
the residue norm on an affinoid subdomain of X coincides with the supremum
norm. The supremum norm is power-multiplicative so the reduction can not
contain any nilpotent elements. Furthermore the reduction is of finite type over
k, so the reduction is a reduced curve. This means the reduction always consists
of a finite union of reduced and irreducible components.
2.2. The formal fiber of a point
Let X/K be an analytic curve with reduction π : X → X̃.
Definition 2.2.1. The preimage of a point x̃ ∈ X̃ under π is called the formal
fiber X+ (x̃) of x̃. An admissible formal affinoid variety U ⊂ X with π −1 (x̃) ⊂ U
is called a formal neighborhood of the formal fiber.
We want to discuss how the kind of the singularity of a point of the reduction
determines the formal fiber.
Proposition 2.2.2. Let X = B1 = Sp Khζi be the unit ball with the canonical
reduction X̃ = A1k = Spec k[Z]. Then {x ∈ B1 ; |ζ(x)| < 1} is the formal fiber
of 0 ∈ X̃. Furthermore O̊X (X+ (x̃)) is local and the reduction map extends to
O̊X (X+ (x̃)) → ÔX̃,x . Furthermore for every f ∈ O̊X we have fg
(x) = f˜(x̃).
Proof. We write x̃ for the zero point of X̃ = A1k . Let x ∈ X+ (x̃) be a point in
the formal fiber with associated maximal ideal mx = (f ) for a function
f=
∞
X
ak ζ k ∈ Rhζi .
k=0
17
2. The structure of a formal analytic curve
Since f˜ generates the ideal (Z) corresponding to the point x̃ we can assume
without restriction that ã0 = 0 and ã1 = 1 and therefore we get |a0 | < 1 and
|a1 | = 1. This means that
∞
1 X
|ζ(x)| = ak ζ(x)k ≤ |a0 | < 1 .
a1
k=0,k6=1
So x is in {x ∈ B1 ; |ζ(x)| < 1}.
On the other hand the function f = ζ − x will reduce to Z for any x in the
given set and therefore (f ) will generate a maximal ideal in the formal fiber.
This means that O̊X (X+ (x̃)) is obtained from
lim O̊X (X)hβ −1 ζi = RJζK .
←−
β<1
P
k
A series f = ∞
k=0 ak ζ is a unit in RJζK if and only if |a0 | = 1, which means
that the sum of two non-units is again not a unit and that O̊X (X+ (x̃)) is local.
Furthermore, for any x ∈ X+ (x̃) we have |f (x)| < 1 if and only if |a0 | < 1
and therefore f˜(x̃) = 0. Since we can use a linear transformation to move any
point into the origin we get fg
(x) = f˜(x̃) for every x ∈ B1 .
Proposition 2.2.3. Let X = Sp A be an analytic curve with smooth reduction
X̃. Then X+ (x̃) is isomorphic to {x ∈ B1 ; |ζ(x)| < 1} for any x̃ ∈ X̃.
Especially O̊X (X+ (x̃)) is local and fg
(x) = f˜(x̃) for every f ∈ O̊X .
Proof. We adapt the proof of [BL85, Prop. 2.2] to our purposes. We choose
a point x ∈ X with π(x) = x̃. Since x and x̃ are regular, we can localize A
in a way that there is a function f with absolute value 1 which generates the
maximal ideal corresponding to x and its reduction will generate the maximal
ideal corresponding to x̃. Now we get
A = K ⊕ Af
and
Ã = k ⊕ Ãf˜ .
And therefore we get
A = K ⊥ Af
(2.1)
according to Proposition 1.1.3. We define the morphism σ : Khζi → A which
maps ζ to f .
Look at

σ:
Khζi

σε :
Khε−1 ζi
_
/ A
_
,
ζ 7→ f
/ Ahε−1 f i
,
ζ 7→ f .
The map σε is injective for all ε since the map f : X → B has zero dimensional
fibers. Furthermore for any element g ∈ Å we set
g0 = g
18
and
gi = gi (x) + gi+1 · f
according to (2.1). So we get |gi | ≤ 1. Then the series
∞
X
i=0
i
gi (x)f =
∞
X
gi (x)εi ε−1 f
i
i=0
converges on Ahε−1 f i and is in the image of σε . But then every series in ε−1 f
is also in the image of σε and therefore σε is surjective.
If we have any g ∈ O̊X such that g̃ ∈ mx̃ we know that g̃ = h̃f˜ and therefore
that |g − h · f | < 1 for any lift h ∈ O̊X of h̃. This means |g(x)| < 1 for any x
with |f (x)| < 1 and so the formal fiber is composed of all x ∈ X which fulfill
|f (x)| < 1.
This gives us the necessary isomorphism of the formal fiber.
With these two propositions, we can describe the formal fiber of a point with
a function f ∈ O̊X . If X̃ is non-singular everywhere but in the interesting point
x̃ and f˜ has a single isolated zero at x̃, we now know that |f (y)| = 1 for any
point outside the formal fiber. So we get {x ∈ X ; |f (x)| < 1} ⊂ X+ (x̃). We
want to prove that equality holds.
Proposition 2.2.4. Let X = Sp B be an analytic curve with reduction X̃. Let
further be f ∈ O̊X (X) be a function so that f˜ has a single isolated zero x̃ and let
X̃ \{x̃} be non-singular. Let further be g̃ ∈ OX̃ 0 a function on the normalization
X̃ 0 of X̃. Then there is an ε < 1 and a lift g ∈ O̊X (Xε ) where Xε = X(|f | ≥ ε).
Proof. Let us first fix some notation. We set A := Khζi and B1K = Sp A. We
can look at f as a function f : X → B1K with a corresponding injection A ,→ B
which maps ζ to f . Let X̃ := Spec B̃, Ak := Spec Ã and X̃ 0 := Spec B̃ 0 , the
latter being the normalization. We will further need the varieties
Xε,s := {x ∈ X ; ε ≤ |f (x)|, |s(x)| = 1}
and
B1ε,s := {x ∈ B1K ; ε ≤ |ζ(x)|, |s(x)| = 1} ,
and their corresponding affinoid algebras Bε,s = Bhεf −1 , s−1 i for Xε,s and
Aε,s = Ahεζ −1 , s−1 i for Bε,s where s is a function in O̊B with s̃(0) 6= 0 chosen
so that f is finite.
Since B0,s is finite over A0,s the quotient field Q(B0,s ) is algebraic over Q(A0,s )
and Q(B̃) = Q(B̃ 0 ) is algebraic over Q(Ã). Furthermore B0,s is the integral
closure of A0,s and B̃s̃0 is the integral closure of Ãs̃ in their respective fields.
Since K is algebraically closed, K is stable. Due to [BGR84, 5.3.2 Thm.
1] Q(Khζi) = Q(A) is stable. Since B/A is generically unramified, we get
according to [BGR84, 3.6 Prop. 8] that
t := [B0,s : A0,s ] = [B̃s̃ : Ãs̃ ] .
(2.2)
The given element g̃ induces a function on X̃ \ {x̃} so there is a lift g ∈
O̊X (X \ X+ (x̃)). The restriction map OX (ε ≤ |f |) → OX (1 ≤ |f |) has a dense
image so we can choose g ∈ OX (ε ≤ |f |) = Bε without restriction.
19
We have already seen that B0,s is integral over A0,s and therefore Bεf −1 ,s is
integral over Aεζ −1 ,s . Completion gives that Bε,s is integral over Aε,s so there
is an integral equation
F (T ) := (−1)t T t + a1 T t−1 + · · · + at
of degree t, that annuls g with ai ∈ Aε,s which is the characteristic polynomial
of g.
Let
X
ai =
aiν ζ ν
ν∈Z
be the Laurent series representation of ai . Because of (2.7) the reduction F̃ of
F is the characteristic polynomial of g̃. Therefore we have ãi ∈ Ãs̃ and thus
|aiν | < 1 for all ν < 0. Furthermore since ai ∈ Aε,s the sequence aiν εν tends to
zero for ν → −∞. So there is an N0 such that |aiν εν | < 1 for all ν < N0 . Since
|aiν | < 1 for all ν < 0 we can find ε1 < 1 such that |aiν εν1 | < 1 for N0 ≤ ν < 0.
Thus, by adjusting ε, we have |aiν εν | < 1 for all ν < 0. Therefore |ai | ≤ 1 and
thus |g| ≤ 1 on Xε,s .
Proposition 2.2.5. Let X, f and g be as in Proposition 2.2.4. Denote the zero
of f˜ by x̃. There is a point x ∈ Xε = X(ε ≤ |f | < 1) such that |g(x)| = 1 if
and only if there is a point ỹ ∈ X̃ 0 lying over x̃ with g̃(ỹ) 6= 0.
Proof. It is important to note that by Proposition 2.2.3 we know that |f (x)| <
1 already implies that x is in the formal fiber. Furthermore we know that
|ak (x)| < 1 for one x with |f (x)| < 1 implies that |ak (x)| < 1 for any x with
|f (x)| < 1 since ak ∈ Aε .
Since x̃ is the only zero of f˜ we know that g̃ has m zeroes on the n points
ỹ1 , . . . , ỹn lying over x̃ if and only if ãk (0) = 0 for k < m and ãm (0) 6= 0. So
|ak (x)| < 1 for k < m and |am (x)| = 1 for any x in the periphery of the formal
fiber. Therefore there is a point x with |g(x)| = 1. The other implication follows
by the same argument.
Lemma 2.2.6. Let X be an analytic curve and x̃ ∈ X̃ a point of the reduction.
Then the ring O̊X (X+ (x̃)) is local with residue field k.
Proof. We can, without restriction, assume that X = Sp B and X̃ = Spec B̃.
We can further assume that X̃ has no singularities beside x̃.
Let h1 , h2 ∈ O̊X (X+ (x̃)) be two non-units. This means that there are points
x1 , x2 ∈ X+ (x̃) such that |h1 (x1 )|, |h2 (x2 )| < 1.
If |h1 (x2 )| < 1 then |(h1 + h2 )(x2 )| < 1 and h1 + h2 is not a unit. Therefore
we can assume that |h1 (x2 )| = 1.
Let us first move back in an affinoid case. Set f = h1 · h2 and choose β < 1
such that |f (x1 )|, |f (x2 )| < β. We set Bβ = Bhf /βi with reduction B̃β . Since
h1 ∈ O̊X (X+ (x̃)) we get h1 ∈ B̊β . By the choice of β the function h1 is still not
a unit in Bβ but |h1 | = 1 in Bβ .
The function h̃1 is defined on B̃β and therefore is defined on the normalization
of X̃β as well. Furthermore, since |h1 (x1 )| < 1 we get h̃1 (x̃) = 0, so h̃1 (ỹ) = 0 for
20
all ỹ lying over x̃ in the normalization. This means |h1 (x2 )| < 1 by Proposition
2.2.5, contrary to our assumption.
Therefore an element of h ∈ O̊X (X+ (x̃)) is not a unit if and only if |h(x)| < 1
on every point x ∈ X+ (x̃). This implies that the sum of two non-units is again
not a unit and O̊X (X+ (x̃)) is local.
Corollary 2.2.7. Let X be an analytic curve and h ∈ O̊X (X+ (x̃)) be a function. If |h(x)| < 1 for any x ∈ X+ (x̃) then |h(x)| < 1 for all x ∈ X+ (x̃).
Corollary 2.2.8. Let X be an analytic curve and f ∈ O̊X be a function such
that f˜ has a single isolated zero in the point x̃. Then X+ (x̃) = X(|f | < 1).
Proof. As we have seen |f (x)| < 1 on every point of X+ (x̃).
Corollary 2.2.9. Evaluating functions and taking their reduction commutes.
We get fg
(x) = f˜(x̃).
Lemma 2.2.10. Let X be a analytic variety and x̃ ∈ X̃ a point of the reduction.
Then O̊X (X+ (x̃)) is henselian.
Proof. Let us set B̊+ := O̊X (X+ (x̃)). Let P = T n +cn−1 T n−1 +· · ·+c0 ∈ B̊+ [T ]
be a monic polynomial with reduction P̃ ∈ k[T ] which has a simple zero α̃ in
k. After coordinate transformation we can assume that α̃ is 0. We denote the
other zeroes by α̃i with i = 2, . . . , n, where n is the degree of P .
In this setup we see that c̃0 = 0 and therefore c0 ∈ m, the maximal ideal of
B̊+ . Since c̃1 is the sum of every possible product of n − 1 zeroes of P̃ and
only one such combination, namely α̃2 · · · · · α̃n differs from zero, we know that
c1 6∈ m.
For every x ∈ X+ (x̃) and |ε| < 1 the polynomial P can be written as
P = εn · T n /εn + cn−1 (x)εn−1 · T n−1 /εn−1 + · · · + c1 (x)εT /ε + c0 (x)
Now c0 ∈ m and c1 6∈ m imply |c0 (x)| = η < 1 and |c1 (x)ε| = |ε| by Corollary 2.2.7. Furthermore for i > 1 we get |ci (x)εi | ≤ |εi | < |ε|, so P is T /ε
distinguished of order 1 if ε > η.
According to the Weierstraß preparation theorem [Bos05, Sect. 1.2 Cor. 9]
this that means there is a unit ωε in KhT /εi and an element αε (x) ∈ K such
that
P (x) = (T /ε − αε (x)) · ωε (x) = (T − αε (x) · ε) · ωε (x)/ε .
(2.3)
Since P is a polynomial and ωε is a convergent power series, ωε can only be a
polynomial itself.
The uniqueness of Weierstraß preparation assures us that α(x) := αε (x) · ε
does not depend on ε and as such gives a function α ∈ B+ . Furthermore,
we know that |T /ε − αε (x)| = 1 in the absolute value of KhT /εi, so |α(x)| =
|αε (x) · ε| ≤ |ε| < 1 which means that α is in m, i.e. α is the lift of α̃ we looked
for.
Corollary 2.2.11. The ring of functions on a formal fiber of a point is invariant under étale base change.
21
2.3. The formal fiber of regular points and double points
Proposition 2.3.1. Let X = Sp A be an analytic curve with reduction X̃. A
point x̃ ∈ X̃ of the reduction is regular if and only if the formal fiber X+ (x̃) is
isomorphic to {x ∈ B1 ; |x| < 1}.
Proof. We have already seen in Proposition 2.2.3 that X+ (x̃) is isomorphic to
the open unit ball if x̃ is regular.
If we suppose that X+ (x̃) is isomorphic to {x ∈ B1 ; |x| < 1} we know by
Lemma 2.2.6 that Ãˆ is isomorphic to kJT K and therefore that x̃ is regular.
Proposition 2.3.2. Let X = Sp A be an analytic curve with reduction X̃. A
point x̃ ∈ X̃ of the reduction is an ordinary double point if and only if the
formal fiber X+ (x̃) is isomorphic to {x ∈ B1 ; ε < |x| < 1}, an open annuli in
B1 .
Proof. At first we want to reduce to the case that the double point lies on two
different components. This can be done by applying an étale base change so
this is simply a consequence of Corollary 2.2.11. However, the same result can
be obtained more directly. This serves to illustrate our proof of said corollary.
Assume the maximal ideal mx̃ corresponding to x̃ is generated by two functions g̃1 , g̃2 such that f˜ := g̃1 · g̃2 ∈ m3x̃ . This is possible since x̃ is a double
point. Assume further that without restriction ordỹi f˜ = 3 on both points ỹi
lying over x̃ in the normalization. This assumption can easily be met by having
ordỹi g̃i = 1 and ordỹi g̃j = 2 for i, j = 1, 2, i 6= j.
Now look at hi := gi3 /f for some lifts gi , f of g̃i , f˜. Their reduction is defined
on the normalization of X̃ and so hi is defined and of absolute value smaller
than 1 on X(|f | ≥ ε3 ) for a suitable ε. But we have h̃i (ỹi ) 6= 0 so there is a
point xi ∈ X+ (x̃) with |hi (xi )| = 1. So we can set β such that |f (xi )| = β 3 < 1
and we get |gi (xi )|3 = |f (xi )| which means that |gi /β| = 1 on X(ε3 ≤ |f | ≤ β 3 ).
Note that by adjusting ε we can get values for β arbitrarily close to 1. This
means that on Ahβ −3 f i we have g1 /β · g2 /β = f /β 2 . But |f | = β 3 on Ahβ −3 f i
so this reduces to g̃10 · g̃20 = 0. Therefore x̃ joins two components of the reduction
of X(|f | ≤ β 3 ) for any β < 1 and we only need to discuss this case.
We will use the proof as in [BL85, Prop. 2.3]. Let mx̃ be generated by f˜
and g̃ such that f˜ · g̃ = 0. Assume that f and g are lifts chosen so that they
have a common zero x. This can be achieved by replacing g with g − g(x) for
a zero x of f . Since |g(x)| < 1 the reduction g̃ is left unchanged. Therefore mx
is generated by f and g and we get
A = K ⊥ Af ⊥ Ag
(2.4)
as in the case of a regular point. We recursively define sequences (fi ), (gi ), (hi )
in A and (αi ) in K. For this set f0 = g0 = α0 = 0 and define
hk :=
f−
k−1
X
i=0
22
!
fi
g−
k−1
X
i=0
!
gi
−
k−1
X
i=0
αi
(2.5)
2.4. The formal fiber of a general singular point
Then we decompose hk according to (2.4) to get
hk = αk + gk f + fk g
(2.6)
Using (2.6) in (2.5) gives us
hk = fk−1
k−2
X
i=0
gi + gk−1
k−2
X
fi + fk−1 gk−1
i=0
for k ≥ 2. By our assumption of f and g we get |h1 | = |f ·g| = γ < 1. Therefore
we recursively get |fk |, |gk |, |αk | ≤ |hk | ≤ γ k . Set
!
!
∞
∞
X
X
f 0 := f −
and
g 0 := g −
fi
gi
i=1
i=1
P
0
˜0
˜
Then f 0 · g 0 = ∞
i=1 αi =: α ∈ K. Since f = f and g̃ = g̃ we know that |c| < 1.
Furthermore c 6= 0 since A is an integral domain.
We define
σ : Khζ, cζ −1 i → A
ζ 7→ f 0
cζ −1 7→ g 0
and again set σε : Khε−1 ζ, ε−1 cζ −1 i → Ahε−1 f 0 , ε−1 g 0 i for the induced map.
We know that σ is injective because
For any element h ∈ A we
P
P of (2.4).
∞
0k with h ∈ K by
0k +
h
f
can construct a series h = h0 + ∞
k
k
k=0 hk g
k=0
repeated application of (2.4). The series converges as long as |ε| < 1 so σε is an
isomorphism in this case.
Theorem 2.4.1. Let X = Sp B be of pure dimension 1, and let x̃ be a point
in X̃. Let U be a formal neighborhood of X+ (x̃) in X and let f be a function
in O̊X (U ) such that x̃ is an isolated zero of f˜ ∈ OX̃ (Ũ ). Let x̃01 , . . . , x̃0n be the
points in the normalization X̃ 0 of X̃ lying over x̃. Then, for ε ∈ |K × |, ε < 1
close to 1, the analytic variety {x ∈ X+ (x̃) ; ε ≤ |f (x)| < 1} decomposes into
n connected components R1 , . . . , Rn which are semi-open annuli.
More precisely, let ζ be a coordinate function of B1 , and, for i = 1, . . . , n,
denote by ti = ordỹi0 (f˜) the vanishing order of f˜ at x̃0i . There are isomorphisms
ϕi : Ri −→{z
˜
∈ B1 ; ε1/ti ≤ |ζ(z)| < 1}
such that, up to a unit in O̊X (Ri ), the function f |Ri equals the pullback ϕ∗i (ζ ti ).
Furthermore, if the reduction h̃ ∈ ÔX̃,x̃ of an element h ∈ O̊X (X+ (x̃)) satisfies τ = ordx̃0i (h̃) < ∞ for some index i0 , and if ε is close enough to 1, then,
0
up to a unit in O̊X (Ri0 ), the function h|Ri0 equals the pullback ϕ∗i0 (ζ τ ).
23
Proof. Let us first provide a proof for the simpler situation where f only has
one isolated zero x̃ in X̃ and only one point x̃0 lies above x̃ in the normalization.
The first step of the proof is to use Lemma 2.2.4 on a local parameter of x̃0 .
Since we have not assumed X̃ \ {x̃} to be non singular, we need to adjust the
proof a little.
Let us recall some notation. Let X = Sp B and B = Sp A with A := Khζi.
We have, as usual, the morphism f : X → B corresponding to an injection
A ,→ B which maps ζ to f . Let X̃ := Spec B̃, Ã := Spec Ã and X̃ 0 := Spec B̃ 0 ,
the latter being the normalization. We will further need the varieties
Xε,s := {x ∈ X ; ε ≤ |f (x)|, |s(x)| = 1}
and
Bε,s := {x ∈ B ; ε ≤ |ζ(x)|, |s(x)| = 1} ,
where s is a function in O̊B and s̃(0) 6= 0 yet to be determined fully but chosen
so that f is finite. Note that Xε,s and Bε,s only depend on ε and the reduction
s̃ of s.
Since K is algebraically closed, K is stable. Due to [BGR84, 5.3.2 Thm.
1] Q(Khζi) = Q(A) is stable. Since B/A is generically unramified, we get
according to [BGR84, 3.6 Prop. 8] that
t := [B0,s : A0,s ] = [B̃s̃ : Ãs̃ ] .
(2.7)
Let g̃ be a local parameter of x̃0 in X̃ 0 . Then we have Q(B̃ 0 ) = Q(Ã)[g̃]. We
can choose s̃ as desired so that localizing at ζ and s̃ yields B̃ζ·s̃ = Ãζ·s̃ [g̃], where
s̃(0) 6= 0.
We can remove singularities by localizing, so we can apply Lemma 2.2.4 to
get an ε such that a lift g of g̃ is defined on Xε,s .
The polynomial F of the lemma is the minimal polynomial of g up to normalization, by our choice of assumptions.
Next, we want to show that
Bε,s = Aε,s [g] .
Since Q(B̃)/Q(Ã) is a vector space, we can adjust s in a way that B̃s̃ is even a
finite free Ãs̃ module, namely
B̃ζ,s̃ = Ãζ,s̃ g̃ 0 ⊕ · · · ⊕ Ãζ,s̃ g̃ t−1 .
By applying Proposition 1.1.3 we get
B1,s = A1,s g 0 ⊥ · · · ⊥ A1,s g t−1
(2.8)
i.e. g 0 , . . . , g t−1 are an orthonormal system of generators of B1,s over A1,s .
Furthermore, since F is a Weierstraß-polynomial, we can decompose A[g]
according to [BGR84, 5.2.3 Prop. 3] so that
A[g] = Ag 0 ⊕ · · · ⊕ Ag t−1
24
and also
Aε,s [g] = Aε,s g 0 ⊕ · · · ⊕ Aε,s g t−1
(2.9)
Since B1,s = A1,s [g] and the support of the quotient Bε,s /Aε,s [g] consists of
finitely many points. So there exists an ε < 1 such that Bε,s = Aε,s [g].
0
Ã(ζ) ,→ B̃(f
) is a finite extension of discrete valuation rings, with ζ̃ being a
local parameter in the former and g̃ in the latter. Therefore there is a unit ũ
so that ζ̃ = g̃ t ũ. Since the residue field is algebraically closed, we can, without
loss of generality, write ũ = 1 + δ̃ with δ̃ ∈ m̃. So we get the representation
ζ = g t (1 + δ)
(2.10)
with |δ(x)| < 1 where for x ∈ Xε,s and |ζ(x)| < 1.
Now δ has a series representation
δ = δ0 g 0 + · · · + δt−1 g t−1
with δi ∈ Aε,s ,
δi =
X
diν ζ ν .
ν∈Z
Due to (2.8) we have |diν | ≤ 1. Due to (2.9) we have diν εν → 0 for ν → −∞
and so there is an N0 such that |diν εν | < 1 for all ν < N0 . We claim |di,ν | ≤ 1
for all ν ∈ Z and |di,ν εν | < 1 for all ν ≤ 0.
We know |δi | ≤ 1 and δi ∈ Aε,s . Since δ̃ ∈ m̃ and
ˆ 0 = Ãg̃
ˆ 0 ⊕ · · · ⊕ Ãg̃
ˆ t−1
B̃
we must have δ̃i ∈ Ã(ζ) ⊂ kJζK. As in the lemma above, we can modify ε if
necessary to get that |diν εν | < 1 for all ν < 0. The case ν = 0 was not covered
above, however, this holds since δ̃ ∈ m̃. This shows that
δ ∈ Rhζ, ε/ζi · g 0 ⊕ · · · ⊕ Rhζ, ε/ζi · g t−1
(2.11)
holds.
On the other hand, by the same reasoning, we also have a γ ∈ Bε,s with
g t = ζ(1 + γ) .
Yet again, we can write
γ = γ0 g 0 + · · · + γt−1 g t−1
with
γi =
X
ci,ν ζ ν
ν∈Z
and the reduction γi ∈ kJζK. So we have |ci,ν | ≤ 1 for all ν ∈ Z and |ci,ν εν | < 1
for ν ≤ 0. Again, this means that
γ ∈ Rhζ, ε/ζi · g 0 ⊕ · · · ⊕ Rhζ, ε/ζi · g t−1
(2.12)
25
Choosing an element α ∈ K with αt = ε, we get
t
ε
α
=
· (1 + γ)
ζ
g
(2.13)
We want to show that for any arbitrary number β with |β| < 1 we have
Rhg/β, α/gi = Rhζ/β t , ε/ζi[g]
and that therefore X+ (x̃) is isomorphic to a semi-open annulus.
Let w ∈ R be a number with |w| = max(|δ|, |γ|) where the absolute value of
the ring Rhζ/β t , ε/ζi[g] is used. Our calculations above have shown that


|w| ≤ max  sup
(di,ν εν ) ,
i=1...t−1
ν≤0
sup
i=1...t−1
ν≤0
(ci,ν εν ) , β t  < 1 .
By (2.10) and (2.13) we have
ζ, ε/ζ ∈ Rhg/β, α/gi + w · Rhζ/β t , ε/ζi[g] .
Since ζ/β t , ε/ζ and g generate Rhζ/β t , ε/ζi[g] this means
Rhζ/β t , ε/ζi[g] = Rhg/β, α/gi + w · Rhζ/β t , ε/ζi[g] .
(2.14)
Applying (2.14) to itself n-times yields
Rhζ/β t , ε/ζi[g] = Rhg/β, α/gi + wn · Rhζ/β t , ε/ζi[g]
And thus, since |w| < 1 and Rhg/β, α/gi is complete, we get
Rhζ/β t , ε/ζi[g] = Rhg/β, α/gi ,
which proves the theorem in this special case.
In the general case, use an étale base change on Ã so that Ãét is an étale
neighborhood of x̃. Let X̃ ét := X̃ ×Ã Ãét and X̃ 0ét := X̃ 0 ×Ã Ãét be the corresponding varieties for X̃. We can chose the étale base change in such way that
X̃ ét consists of n components which decompose in X̃ 0ét in n disjoint components, where n is the number of points in the normalization X̃ 0 over x̃. In this
case X̃ ét \ {x̃} also decomposes into n disjoint components.
ét (x̃) →
Let X ét be a lifting of X̃ ét . In particular we have X+
˜ X+ (x̃) by Corolét
lary 2.2.11. Since X̃ \ {x̃} consists of n disjoint components, X1 := {x ∈
X ét ; |f (x)| = 1} consists of n disjoint components as well. Now look at
Xε0 := {x ∈ X ét ; |f (x)| ≥ ε0 } and consider a function h ∈ OX (X1 ) which
behaves like a constant on any connected component with pairwise different
absolute values unequal to zero on the components of X1 . Since the image of
OX (Xε0 ) is dense in OX (X1 ), we can choose a function ĥ ∈ OX (Xε0 ) which
approximates h such that |h − ĥ|X1 | < min{|h(x)| ; x ∈ X1 }. The function
|ĥ| : Xε0 → R is continuous with regard to the G-topology on Xε0 . Since the
absolute values on the components of X1 are different, we can choose a δ for
26
2.5. Formal blow-ups
i 6= j so that ||h|i − |h|j | > 2δ > 0, where |·|i is the absolute value on the
i-th component of X1 . Since |ĥ| is continuous, there
is an ε0 so that for every
x ∈ Xε0 we get an y ∈ X1 so that |ĥ(x)| − |h(y)| < δ. By our choice of δ this
is only possible if Xε0 also consists of n disjoint components.
Now look at the i-th component X̃i of X̃ ét . Similar to the special case, choose
g̃i ∈ OX̃ 0 (X̃i0 ) as a local parameter of x̃0i . We can extend this function to X̃ 0ét
by zero on the other components. Now lift g̃i to gi on X1 . As in the special case
we see that, for εi > ε0 , we can chose this lift in OX (Xεi ). As above, gi will,
after adjusting εi , generate the algebra of {x ∈ Xi ; ε ≤ |f (x)| < 1}. We get, as
in the special case, that the i-th component Xi of Xε0 will satisfy the theorem,
e.g. that {x ∈ Xi ; εi ≤ |f (x)| < 1} is isomorphic to a semi-open annulus with
local parameter gi . Since gi vanishes on all the other components, choosing an
ε ≥ maxi=0,...,n εi , we get that {x ∈ X ét ; ε ≤ |f (x)| < 1} is isomorphic to a
disjoint union of semi-open annuli.
Since the base change is isomorphic on the formal fiber the theorem is also
proved in the general case.
For the additional assertion of the theorem, let h ∈ O̊X (X+ (x̃)) satisfy τ =
ordx̃0i (h̃) < ∞ for some index i0 . Since the base change does not change the
0
fact that g̃i0 is a local parameter of the reduction of the annulus Ri0 we have
the equation
h̃ = ũ · g̃iτ0 .
By lifting this equation we see that, up to a unit, h|Ri0 equals giτ0 , the pullback
of the coordinate of Ri0 .
2.5. Formal blow-ups
In this section we want to define the concept of a formal blow-up.
Definition 2.5.1. Let X be an analytic curve with reduction π : X → X̃
corresponding to the admissible formal covering U = (Ui )i∈I . Let f ∈ O̊X (Ui )
be a function such that f˜ has an isolated zero x̃ in Ũi so that x̃ 6∈ Ũj for any
j 6= i and ε ∈ K with |ε| < 1. The formal blow-up corresponding to f and ε is
the admissible formal covering U \ Ui ∪ {Ui (|f | ≤ ε), Ui (|f | ≥ ε)}.
Proposition 2.5.2. Let X be an analytic curve with formal covering U. Let
Û denote the formal blow up of U according to Ui , f and ε ∈ |K × | as defined
ˆ → X̃ which
above. Then Û is formal. The formal blow up induces a map ϕ̃ : X̃
is an isomorphism on X̃ \ {x̃} and is the identity on X.
Proof. The covering Û is admissible because Ui (|f | ≤ ε) and Ui (|f | ≥ ε) are
rational subdomains and as such admissible open. We have |f | = ε on Ui0 :=
Ui (|f | ≤ ε). This means that f /ε is in Ůi (|f | ≤ ε) and the reduction of Ui (|f | =
ε) equals the set Ũ 0 g . Similarly Ui (|f | = ε) reduces to Ũ 00g on Ui00 := Ui (|f | ≥
i,f /ε
i,ε/f
ε).
The identity on X will map every point of Ũi (|f | ≤ ε) to the point x̃. On
Ũi (|f | ≥ ε) every lift y with |f (y)| < ε of a point ỹ is in the formal fiber of x̃
and as such ỹ will be mapped on x̃. Every other point is left unchanged.
27
Proposition 2.5.3. Let X be an analytic curve with formal covering U. Let
x̃ ∈ X̃ be a point in the associated reduction. There is a formal covering Û
which induces the same reduction X̃ such that x̃ ∈ Ũi for exactly one i.
Proof. For every Ui with x̃ ∈ Ũi we can find an f˜i ∈ OX̃ (Ũi ) which does not
vanish except in isolated points and that these points lie in another set Ũj as
well and f˜i (x̃) = 0. Moreover we can assume that f˜i and f˜j have only x̃ as
common zero on Ũi ∩ Ũj . Replacing Ui for all but one affected index with
Ui (|fi | = 1) where fi is any lift of f˜i does not change the reduction associated
to U.
Definition 2.5.4. Let X be a proper analytic curve with fixed reduction X̃
and let x̃ ∈ X̃ be a point. Let f and ε < 1 be as in Theorem 2.4.1 so that
{x ∈ X+ (x̃) ; ε ≤ |f (x)| < 1} consists of n disjoint semi-open annuli. We
define the proper curve X x̃ by pasting n discs B1 into the formal fiber X+ (x̃)
via these annuli.
To be more precise, let
ϕi : Ri −→{z
˜
∈ B1 ; ε1/ti ≤ |ζ(z)| < 1}
be the maps of Theorem 2.4.1 where Ri is the ith connected component of
{x ∈ X ; ε ≤ |f (x)| < 1} and let Bi = B1 together with
˜
∈ B1 ; ε1/ti ≤ |ζ(z)| < 1} ; x 7−→ ε1/ti /x
ψi : Bi (|x| > ε1/ti ) −→{z
Then X x̃ is defined by X+ (x̃), B1 , . . . , Bn and the gluing relations ψi−1 ◦ ϕi .
Proposition 2.5.5. There is an canonical admissible formal covering U =
(Ui )ni=0 of X x̃ . The associated reduction X̃ x̃ consists of n components which
are rational curves joined by a single singular point x̃ with the same type of
singularity as x̃ in X̃.
Proof. By [BL85, Prop. 4.1.] the covering consisting of U0 = X x̃ \(B1+ ∪· · ·∪Bn+ )
and Ui = Bi and Bi+ as the open unit ball is a formal covering. The associated
reduction has n affine lines derived from the Bi glued together with a single
x̃ (x̃). So the type of singularity is the same according
point x̃ with X+ (x̃) = X+
to Proposition 2.2.6.
Definition 2.5.6. We define g(x̃) = g(X x̃ ) to be the genus of X x̃ . We further
set n(x̃) as the number of connected components of X x̃ \ X+ (x̃).
Lemma 2.5.7. If g(x̃) = 0 and n(x̃) = 1 then x̃ is a regular point. If g(x̃) = 0
and n(x̃) = 2 then x̃ is an ordinary double point.
Proof. If g(X x̃ ) = 0 we know that X x̃ is a rational curve and as such isomorphic
to P1 . Therefore we know that X+ (x̃) is isomorphic to P1 \ B1 if n(x̃) = 1 and
˙ 1 ) if n(x̃) = 2. In the first case we get an open disc, in the second
P1 \ (B1 ∪B
an open annulus, proving our assertion by Proposition 2.3.1 and Proposition
2.3.2.
28
2.6. The stable reduction theorem
Definition 2.5.8. Let X̃ be a projective connected curve over k consisting of
n components. The cyclomatic number z(X̃) is the number
X
z(X̃) =
(n(x̃) − 1) − n + 1
x̃∈X̃
where n(x̃) is again the number of points lying over x̃ in the normalization.
Note. Since an ordinary double point has n(x̃) = 2, this definition coincides
with the one we give in Chapter 4 where the cyclomatic number is defined as
the number of double points minus the number of components plus one.
The main missing part of the semi-stable reduction theorem is the genus
formula. We do not give a proof for this, and just state the results here.
Theorem 2.5.9. We get
g(X) =
n
X
i=1
g(X̃i0 ) +
X
g(x̃) + z(X̃)
x̃∈X̃
Proof. [BL85, Section 4]
With this formula, one can relate the genus of X x̃ with the genus of points
generated in a formal blow-up. If one chooses the function f correctly, one get
the following lemma, which guarantees, that the singularities will always get
better by a formal blow-up.
Lemma 2.5.10. Let K be an algebraically closed non-Archimedean field. Let
X be a projective analytic curve over K with g(X) ≥ 1. There is a meromorphic function f and an ε ∈ |K × | such that the formal covering given by
{x ∈ X ; |f (x)| ≤ ε} and {x ∈ X ; |f (x)| ≥ ε} fulfills g(x̃) < g(X) for all
x̃ ∈ X̃.
Proof. See [BL85, Lemma 7.2]
2.6. The stable reduction theorem
Definition 2.6.1. A projective, connected, reduced curve X̃/k that has only
ordinary double points as singularities is called semi-stable. A semi-stable curve
X̃/k is stable if moreover every component that is isomorphic to P1k meets the
rest of X̃ in at least three points and the arithmetic genus of X̃ is at least two.
Theorem 2.6.2. Let X be a projective analytic curve with reduction X̃ over a
ˆ
field K which is algebraically closed and complete. Then there is a reduction X̃
generated from X̃ by a finite amount of formal blow ups, that has semi-stable
reduction.
x̃ (x̃) we can apply 2.5.10 to every point of X̃ which is
Proof. Since X+ (x̃) = X+
not regular or an ordinary n-fold point. By Induction it follows that every point
is an ordinary n-fold point after a finite amount of formal blow-ups. Ordinary
n-fold points can be broken down further by the methods of [BL85, Chapter 5]
to get to ordinary double points which proves the assertion.
29
Proposition 2.6.3. Let X be a projective rigid analytic curve with reduction
X̃.
Sk Let X̃1 , . . . , X̃k be a selection of irreducible components of X̃ such that0
associated reduction X̃
i=1 X̃i 6= X̃. There is a formal covering U of X with
Sk
and a finite number of point x̃1 , . . . , x̃k such that X̃ \ i=1 X̃i = X̃ 0 \{x̃1 , . . . , x̃k }.
Proof. According to Riemann–Roch
there is a rational function f˜ such that
Sk
˜
f (x̃) = 0 if and only if x̃ ∈ i=1 X̃i which is not constant on every other
irreducible component. A meromorphic lift f of f˜ gives the formal covering
consisting of U1 = {x ∈ X ; |f (x)| ≤ 1} and U2 = {x ∈ X ; |f (x)| ≥ 1}. These
sets are affinoid according to Proposition 1.10.4 and the covering is formal since
the intersection {x ∈ X ; |f (x)| = 1} is an affinoid subvariety of both U1 and
U2 .
Take a point x̃ ∈ X̃i with i > k i.e. f˜ is not constant on X̃i . We can assume
without restriction that f˜(x̃) = c̃ for c ∈ K. The formal fiber of x̃ can be
written as X+ (x̃) = {x ∈ U ; |f (x) − c| < 1} where U is the preimage of an
affine neighborhood Ũ of x̃ such that x̃ is the only point in Ũ with f˜(x̃) = c̃. But
{x ∈ U ; |f (x) − c| < 1} is a formal fiber in {x ∈ X ; |f (x)| = 1} as well, so the
identity
Sk on X reduces to an isomorphism on these components. All the points
of i=1 X̃i will be mapped on a connected component of {x ∈ X ; |f (x)| < 1}
which is a finite disjoint union of formal fibers in {x ∈ X ; |f (x)| ≤ 1}. So the
components are mapped to a finite set of isolated points in X̃ 0 .
Definition 2.6.4. The reduction X̃ 0 of Proposition 2.6.3 is called the blowdown of the components X̃1 , . . . , X̃k in X̃.
Theorem 2.6.5. Let X be a projective analytic curve of genus at least two and
with semi-stable reduction X̃, associated to the formal covering U. There is a
formal covering Û of X inducing stable reduction.
Proof. Let X̃i be a rational component of X̃ which intersects the rest of the
curve in only two ordinary double points. Since the genus of X is at least
two, we know that X̃i is not the only irreducible component of X̃. This means
X̃i \Sing X̃ is isomorphic to P1 missing two discs. But then Xi := π −1 (X̃i \ X̃) is
isomorphic to Khζ, ζ −1 i with ζ defined on the formal fiber of the double points.
In fact ζ is a coordinate of both these open annuli.
We blow down this component using Proposition 2.6.3. This maps the whole
component to a single point x̃ ∈ X̃ 0 . The formal fiber of this point is the union
of Sp Khζ, ζ −1 i with two open annuli, each of which having ζ as coordinate.
This results in an open annulus, again with ζ as coordinate, so x̃ is an ordinary
double point according to Proposition 2.3.2. Therefore X̃ 0 is again semi-stable,
but misses the component X̃i of X̃. Repeating this process until all rational
components with only two intersections with the rest of the curve are gone
yields a stable reduction.
30
2.7. Examples
2.7. Examples
Let us show the methods of this chapter in the example of hyperelliptic curves.
Let X be defined by
U1 := Sp Khξ, ηi/(η 2 − ξ(ξ − λ2 ) . . . (ξ − λn ))
U2 := Sp Khσ, τ i/(τ 2 − σ(1 − σ)(1 − λ1 σ) . . . (1 − λn σ))
with n = 2g − 1 and gluing relations ξ = 1/σ and η = τ /σ n+1 . We can assume
without restriction that |λi | ≤ 1 by application of an affine map. This covering
is formal with associated reduction X̃ given by
Ũ1 := Spec k[X, Y ]/(Y 2 − X(X − 1)(X − λ̃1 ) . . . (X − λ̃n ))
Ũ2 := Spec k[S, T ]/(T 2 − S(1 − S)(1 − λ̃1 S) . . . (1 − λ̃n S))
and corresponding gluing relations. The curve X̃ is smooth if λ̃i 6= λ̃j for i 6= j
or equivalently if |λi − λj | = 1 for i 6= j.
Otherwise we can move a singularity to the point zero so that we can assume
without restriction that the function f˜ = X has a single isolated zero in the
offending singularity on Ũ1 . We choose f = ξ as a lift. Assume that the λi are
sorted by absolute value and that |λi | < 1 for i ≤ k and |λi | = 1 for i > k. Set
ε = max |λi | .
1<i≤k
There is no branching point in {x ∈ U1 ; ε < |ξ(x)| < 1}. So this set is composed of one annulus if k is odd and two disjoint annuli if k is even with a
coordinate calculated by the covering morphism η. Blowing U1 up with parameters f and ε gives
U10 := Sp Khξ, η, ζi/(ξ − εζ, η 2 − ξ(ξ − λ2 ) . . . (ξ − λn ))
U100 := Sp Khξ, η, ζi/(ζξ − ε, η 2 − ξ(ξ − λ2 ) . . . (ξ − λn ))
After restructuring the ideals we get
U10 := Sp Khξ, η 0 , ζi/ ξ − εζ, η 02 − ζ(ζ − λ2 ε−1 ) . . . (ζ − λk ε−1 )·
(ξ − λk+1 ) . . . (ξ − λn ))
U100
00
:= Sp Khξ, η , ζi/ ζξ − ε, η
002
− ζ j (1 − λ2 ε−1 ζ) . . . (1 − λk ε−1 ζ)·
(ξ − λk+1 ) . . . (ξ − λn ))
where j = 0 if k is even and j = 1 otherwise. The new variable η 0 is just η
modified by a constant and η 00 is η/ξ dk/2e also modified by a constant. This
gives the reductions
^
−1
−1
Ũ10 := Spec k[Y 0 , Z]/ Y 02 − Z(Z − λ^
2 ε ) . . . (Z − λk ε )
^
−1
−1
Ũ100 := Spec k[X, Y 00 , Z]/ XZ, Y 002 − Z j (1 − λ^
2 ε Z) . . . (1 − λk ε Z)·
(X − λ̃k+1 ) . . . (X − λ̃n )
31
We see that Ũ100 consists of two irreducible components, both hyperelliptic curves
joint together in a single double point if j = 1 or two ordinary double points if
j = 0. The variety Ũ10 just closes the new curve.
The remaining singularities can be handled by the same process. Therefore
the stable reduction of a hyperelliptic curve consists of hyperelliptic curves
joined together by ordinary double points.
32
In this chapter we want to give an overview of the theory of group objects and
Jacobian varieties, which we will need in the next chapter.
3.1. Some definitions from category theory
Definition 3.1.1. A locally small category C is a category in which Hom(X, Y )
is a set for any object X, Y ∈ C.
Definition 3.1.2. An object I ∈ C is called initial if the set Hom(I, X) contains
exactly one element for any object X ∈ C. Dually an object T is called terminal
if Hom(X, T ) contains exactly one element. An object that is both initial and
terminal is called zero object of C
Proposition 3.1.3. Initial, terminal and zero objects of a category are unique
up to isomorphism.
The unique morphism in Hom(I, X) and Hom(X, T ) is denoted by 0. If C has
a zero object then 0 ∈ Hom(X, Y ) denotes the unique morphism that factors
through the zero object.
Definition 3.1.4. Let C be a category with a zero object. For any morphism
ϕ ∈ Hom(X, Y ) we call k ∈ Hom(K, X) the kernel of ϕ if ϕ ◦ k = 0 and if there
is for any t ∈ Hom(T, X) with ϕ ◦ t = 0 a unique morphism u ∈ Hom(T, K)
such that t = k ◦ u. In other words we have the universal property
0
k
KO
u
t
/X
>
ϕ
/7 & Y
0
T
Dually we call c ∈ Hom(Y, C) the cokernel of ϕ if the following universal
property
0
X
ϕ
/Y
0
/& C
c
t
' u
T
holds.
33
Definition 3.1.5. Let X, Y, S ∈ C be objects with morphisms ϕ ∈ Hom(X, S)
and ψ ∈ Hom(Y, S). The pullback or fibered product of ϕ and ψ is an object
P ∈ C together with morphisms p1 ∈ Hom(P, X), p2 ∈ Hom(P, Y ) such that
the following universal property
T
t1
u
t2
P
p1
p2
~
X
Y
ϕ
ψ
S

holds.
Dually an object P with morphisms i1 ∈ Hom(X, P ), i2 ∈ Hom(Y, P ) is called
the pushout of ϕ ∈ Hom(S, X), ψ ∈ Hom(S, Y ) if it adheres to the universal
property
A TO ]
u
t1
i1
X`
t2
>P `
ϕ
i2
ψ
S
>Y
.
If
`ϕ and ψ are clear from the context we write X ×S Y for the pullback and
X S Y for the pushout. The pullback relative to the terminal object is called
the product of X and Y and the pushout relative to the initial object is called
the coproduct of X and Y . If t1 and t2 are
` the morphisms of the diagram we
write t1 × t2 : T → X ×S Y or (t1 , t2 ) : X S Y → T for the respective unique
morphism of the universal property.
If x : T → X and y : T → Y as well as f : X × Y → Z are morphisms we
write f (x, y) for the morphism f ◦ (x × y).
Definition 3.1.6. A locally small category C is called additive if it has a zero
object, all finite products exist and Hom(X, Y ) has the structure of an abelian
group, compatible with concatenation of arrows.
Proposition
` 3.1.7. An additive category has all finite coproducts and the coproduct X Y is canonically isomorphic to X × Y for all objects X and Y .
Proof. We only need to show that X × Y is the coproduct of X and Y . We set
i1 = idX ×0 and i2 = 0 × idY . For any t1 : X → T, t2 : Y → T we set
u = p1 ◦ t1 + p2 ◦ t2 : X × Y → T .
34
3.2. Group objects
We then have
ij ◦ u = ij ◦ (p1 ◦ t1 + p2 ◦ t2 )
= ij ◦ p1 ◦ t1 + ij ◦ p2 ◦ t2
= tj
for j = 1, 2. Note that one needs the commutativity for u to be well defined
and unique.
Definition 3.1.8. An additive category C is called pre-abelian if it contains
every kernel and cokernel.
Definition 3.1.9. A pre-abelian category is called abelian if every monomorphism is the kernel and every epimorphism is the cokernel of some morphism.
3.2. Group objects
Let C be a locally small category with a terminal object 0 and all pullbacks.
Let us also write 0 for the terminal morphism and pi for the ith projection of
the product. Likewise, we will omit ◦ if no confusion is possible.
Definition 3.2.1. Let C be a category as above. We write PC for the category
with the objects (X, e : 0 → X) where X is an object in C and e is a section of
0 and the morphisms of C which respect e, i.e.
HomPC (X, Y ) = {ϕ ∈ HomC (X, Y ) ; eY = ϕ ◦ eX } .
Remark. The category PC has 0 as the zero object and every morphism f : X →
Y has the kernel p1 : X ×Y 0 → X as the universal property of the pullback is
just the universal property of this kernel.
Definition 3.2.2. A group object G in C is an object G together with three
morphisms e : 0 → G, m : G × G → G, i : G → G such that
1. m◦((m◦p12 )×p3 ) = (m◦(p1 ×(m◦p23 )))◦α with respect to the canonical
isomorphism α : (G × G) × G →
˜ G × (G × G),
2. m ◦ (idG ×(e ◦ 0)) = m ◦ ((e ◦ 0) × idG ) = idG ,
3. m ◦ (idG ×i) = m ◦ (i × idG ) = e ◦ 0.
A group object is called commutative if moreover
4. m = m ◦ σ holds for σ := p2 × p1 : G × G → G × G.
Note. To clarify notation, we will use f + g : T → G or f · g for m(f, g) where
f, g : T → G are morphisms. Likewise, we use −f or f −1 for i ◦ f and 0 or 1 for
the morphism e ◦ 0. When dealing with two different group objects, we will use
additive notation for one and multiplicative notation for the other to further
clarify where the operation takes place. If the group object is commutative we
use fractions for a compact notation of the inverse of the multiplicative notation.
35
Note that groups are just the group objects of the category of sets, which we
call abstract groups. Many elementary properties of abstract groups carry over
to group objects. Since e is a section of 0, we know that 0 is epic and can use
the usual proofs to show that e and i are uniquely determined by the morphism
m.
Group objects of a category together with the morphisms compatible with
m, i and e form a category themselves which we denote by GC . We call these
morphisms (group)homomorphisms and will specify the category if confusion
is possible. As in the case of abstract groups the compatibility with m induces
those for e and i and we get
HomGC (X, Y ) = {ϕ ∈ HomC (X, Y ) ; mY (ϕ, ϕ) = ϕ ◦ mX }
One checks easily that this category has 0 as zero object and the pullbacks
A ×C B has canonical morphisms m := mA (p11 , p21 ) × mB (p12 , p22 ), inducing i
and e making it the pullback in GC .
Definition 3.2.3. We have a forgetful functor GC → Grp which associates the
group of abstract points Hom(0, G) to a group object G. For any S ∈ C we can
equivalently define a forgetful functor which associates the group Hom(S, G) to
the S-valued points G(S) of G.
The commutative group objects form a subcategory of GC which we write as
ZC . In this subcategory we see that A × B forms a coproduct of the group
objects A and B. For any object T and a commutative group object G in C
we can induce an abelian group structure on Hom(T, G), so ZC is an additive
category. Both GC and ZC are a subcategory of PC .
Lemma 3.2.4. Every morphism ϕ : X → Y in GC has a kernel and this kernel
is equal to the kernel of ϕ in PC .
Proof. As said above X ×Y 0 has a group structure.
In many cases the category ZC will have cokernels. Thereby ZC is a preabelian category.
Definition 3.2.5. Assume that all cokernels in ZC exist. For a homomorphism
ϕ we define the image of a morphism as im ϕ := ker coker ϕ and dually the
coimage as coim ϕ := coker ker ϕ.
We say that a series of morphisms ϕi : Gi → Gi+1 forms an exact sequence if
im ϕi = ker ϕi+1 . We say that an exact sequence is strict exact if the canonical
morphism coim ϕi → im ϕi is an isomorphism for any i.
Note that the assertion of coim ϕ being isomorphic to im ϕ is automatically
fulfilled in an abelian category. However, in the cases we are interested in, ZC
is not abelian.
Proposition 3.2.6. Let ZC have all cokernels. For any morphism ϕ we have
ker coker ker ϕ = ker ϕ and coker ker coker ϕ = coker ϕ.
36
3.3. Central extensions of group objects
Proof. Look at the diagram
k
KO
/G
>
x
v
ϕ
/T
O
c
X
u
C
where k is the kernel of ϕ and c is the cokernel of k, while x is arbitrary such
that c ◦ x = 0. Since c is the cokernel of k and ϕ ◦ k = 0 there is a unique arrow
u : C → T such that ϕ = u ◦ c. So we get 0 = u ◦ 0 = u ◦ c ◦ x = ϕ ◦ x. But k
is the kernel of ϕ, so there is a unique arrow v : X → K making the diagram
commutative which just means that k is the kernel of c, as required.
The other part of the assumption follows by the dual diagram.
Therefore a short sequence is strict exact if the first morphism is the kernel
of the second and the second morphism is the cokernel of the first.
Definition 3.2.7. Let 0 → G → E → B → 0 be a strict exact sequence in ZC .
We call E an extension of B by G. Two extensions E and E 0 are isomorphic if
there is an isomorphism f such that the diagram
0
/G
id
0
/G
/E
f
/ E0
/B
/0
(3.1)
id
/B
/0
commutes. We denote by Ext(B, G) the class of extensions of B by G up to
isomorphism.
We want to classify these group extensions. For abstract commutative groups
this is done by using the notion of the central extension and group cohomology.
The same process can, with some restrictions, be applied to commutative group
objects which we will sketch in the next theorem.
For a strict exact sequence 0 → G → E → B → 0, we use multiplicative
notation on G and additive notation on E and B. Since all group objects are
commutative, we use fractions for the inverse in multiplicative notation.
Definition 3.3.1. Let B and G be commutative group objects. We call the
set of morphisms f ∈ Hom(B 2 , G) satisfying
f (y, z) · f (x, y + z)
=1
f (x + y, z) · f (x, y)
(3.2)
for the projections x, y, z of B × B × B the cocycles Z 2 (B, G) of B with coefficients in G. The coboundries B 2 (B, G) are morphisms given by
f (x, y) =
g(x) · g(y)
g(x + y)
(3.3)
37
where g : B → G is an arbitrary morphism and x, y are the projections of B ×B.
The second cohomology group of B with coefficients in G is given by
H 2 (B, G) = Z 2 (B, G)/B 2 (B, G) .
The restriction of Z 2 (B, G) to symmetric morphisms f : B 2 → G where
f (x, y) = f (y, x) is denoted by Z 2 (B, G)s . We set
H 2 (B, G)s = Z 2 (B, G)s /B 2 (B, G) .
Theorem 3.3.2. Let C be a category as in Section 3.2 such that ZC has all
cokernels. Let B and G be objects in ZC . The class of commutative extensions
0
/G
ψ
/Ei ρ /B
/0
s
of B by G up to isomorphism which admit a section of ρ in the category C is
in one-to-one correspondence to the group H 2 (B, G)s .
Proof. Take a symmetric cocylce f ∈ H 2 (B, G)s . We can assume without
restriction, that f (eB , eB ) = eG because we can otherwise modify f with the
coboundry induced by g = f ◦∆, where ∆ is the diagonal morphism. We endow
E := G × B with a group law by setting
e := eG × eB
m := (g1 · g2 · f (b1 , b2 )) × (b1 + b2 )
1
i :=
× (−b)
g · f (b, −b)
ρ := p2
ψ := idG ×eB
s := eG × idB
where g1 , g2 , b1 , b2 , g, b are the projections of (G × B) × (G × B) and G × B
respectively.
Using the cocycle condition (3.2) for idB ×eB × eB gives
f (eB , eB ) = f (eB , b) = eG
so e is neutral and i is an inverse. The associativity follows directly by the
cocycle condition (3.2) and the commutativity is implied by f being symmetric.
One checks directly that ψ and ρ are indeed homomorphisms in ZC . Let
X ∈ ZC be any object and ϕ : E → X a grouphomomorphism with ϕ ◦ ψ = 0
so we get the following diagram
G
ψ
/E
0
ρ
/B
ϕ
' u
X
38
We set u = ϕ ◦ s. Let b1 and b2 be the projections of B × B then
u ◦ b1 + u ◦ b2 = ϕ ◦ s ◦ b1 + ϕ ◦ s ◦ b2
= ϕ ◦ (s ◦ b1 + s ◦ b2 )
= ϕ ◦ (1 × b1 + 1 × b2 )
= ϕ ◦ (f (b1 , b2 ) × (b1 + b2 )) .
We know that ϕ ◦ ψ ◦ (f (b1 , b2 ))−1 = 0 so we get
ϕ ◦ (f (b1 , b2 ) × (b1 + b2 )) = ϕ ◦ (1 × (b1 + b2 )) = ϕ ◦ s ◦ (b1 + b2 )) = u(b1 + b2 ) ,
which means that u is a homomorphism, which is automatically unique since ρ
is epic. By the same argument we get
ϕ = ϕ(p1 , ρ) = ϕ(1, ρ) = ϕ ◦ s ◦ ρ = u ◦ ρ .
We also have
ker ρ = E ×B 0 = (G × B) ×B 0 ,
which is isomorphic to G via ψ by the universal properties of product and
pullback in C. Since the kernels in PC and ZC agree, we know that ψ = ker ρ.
So ψ and ρ form a strict exact sequence.
On the other hand, for a given strict exact sequence
0
/G
ψ
/Ei ρ /B
/0
s
with a section s of ρ we can assume without restriction that s ◦ eB = eE , since
otherwise we can replace s by s − s ◦ eB .
We define r0 : E → E ; r0 = idE −s ◦ ρ, so ρ ◦ r0 = eB which implies that
there is a morphism r : E → G ; ψ ◦ r = r0 since ψ = ker ρ in PC as well. But
r0 ◦ ψ = ψ, so r is a retraction of ψ.
Thus we get an isomorphism ϕ : G×B → E by ϕ = ψ◦g+s◦b and ϕ−1 = r×ρ.
We set f 0 := s ◦ b1 + s ◦ b2 − s ◦ (b1 + b2 ). Then ρ ◦ f 0 = eB so there is f such
that f 0 = ψ ◦ f . We can define a group law on G × B via ϕ as
m : (G × B) × (G × B) → G × B
m = ϕ−1 ◦ mE ◦ (ϕ ◦ p1 × ϕ ◦ p2 )
= ϕ−1 ◦ (ψ ◦ g1 + s ◦ b1 + ψ ◦ g2 + s ◦ b2 )
= (g1 · g2 · f (b1 , b2 )) × (b1 + b2 ) .
For another section s0 of ρ we can define g : B → G such that ψ ◦ g = s0 − s
since ρ ◦ (s0 − s) = eB . The cocycles corresponding to s and s0 just differ by
the coboundry induced by g. On the other hand any morphism g : B → G can
be induced this way by setting s0 = s + ψ ◦ g. By the definition of isomorphy
of extensions, this shows that two extensions are isomorphic if and only if they
induce the same cohomology class in H 2 (B, G)s .
39
Definition 3.3.3. Let α : G → G0 be a homomorphism in ZC and let
1
/G
ψ
/Ei ρ /B
/1
s
α
G0
be a strict exact sequence admitting a section. Then the extension α∗ E ∈
Ext(B, G0 ) corresponding to α∗ f := α ◦ f , where f is the cocycle describing E,
is called the pushout of α.
Proposition 3.3.4. The pushout of a homomorphism α is the categorical theoretic pushout of α and ψ. It is the unique extension of B by G0 such that a
homomorphism A exists such that the diagram
1
/G
α
1
ψ
/E
A
/B
id
/B
/ E0
/ G0
/1
/1
commutes.
Proof. The pushout of α and ψ is given by the cokernel of −α×ψ. This cokernel
is just p1 + α ◦ r × ρ ◦ p2 . We set A := α ◦ p1 × p2 which makes the diagram
commutative. The uniqueness of E 0 follows by the uniqueness of the categorical
pushout.
Definition 3.3.5. Let γ : B 0 → B be a homomorphism in ZC and
B0
1
/G
ψ
/Ei
ρ
γ
/B
/1
s
a strict exact sequence admitting a section. The extension γ ∗ E of B 0 by G
given through the cocycle γ ∗ f := f ◦ (γ ◦ p1 ) × (γ ◦ p2 ) where f is the cocycle
corresponding to E is called the pullback of γ.
Proposition 3.3.6. The pullback of a homomorphism γ is the categorical theoretic pullback of γ and ρ. It is the unique extension of B 0 by G such that a
homomorphism Γ exists such that the diagram
1
/G
id
1
/G
/ E0
/ B0
Γ
γ
/E
ρ
/B
/1
/1
commutes.
Proof. Since the categorical pullbacks of ZC and PC coincide we get
E 0 = (G × B) ×B B 0 →
˜ G × B0
as underlying object for the pullback in ZC . Using this canonical isomorphism
to induce a group law produces the given central extension.
40
3.4. Algebraic and formal analytic groups
We are interested in group objects of the category of algebraic varieties, i.e.
separated, connected and integral schemes of finite type over an algebraically
closed field which we call algebraic groups and the group objects of the category
of connected, quasi-separated, quasi-paracompact rigid K-spaces, which we will
call analytic groups. We are especially interested in analytic groups with a fixed
reduction π : X → X̃, where X̃ is an algebraic group, on which the group laws
are given as the reduction of the group laws on X, in other words a formal
group scheme over R.
In the rest of this section we will collect some statements about algebraic
groups and look to generalize them to their analytic counterparts.
Proposition 3.4.1. Algebraic and analytic groups are non-singular.
Proof. Let G be the group in question and τx : G → G be the left translation
by a closed point x ∈ G. This induces an isomorphism of the tangent space
of G at 0 and at x. Since we defined algebraic and analytic groups as reduced
schemes this proves the assumption.
Proposition 3.4.2. Let G and H be algebraic groups and U ⊂ G a non empty
open subvariety. Let ϕ : U → H be a morphism compatible with the group law.
Then there is a unique group homomorphism ϕ̂ : G → H which restricts to ϕ.
Proof. Let x ∈ G be any closed point. Then U − x is isomorphic to U and
open. G is integral and therefore irreducible, so U ∩ (x − U ) is non empty.
Take a = x − b from that intersection and set ϕ̂(x) = ϕ(a) + ϕ(b). Since
ϕ is compatible with the group law the choice of a does not matter and ϕ̂
restricts to ϕ. Furthermore for any fixed a ∈ U we can set ϕa : U + a → H by
ϕa (x) = ϕ(x − a) + ϕ(a) and obtain a morphism that coincides with ϕ on the
common intersection. So ϕ̂ is a morphism of algebraic varieties and respects
the group law. It is unique since rational functions are uniquely determined by
an open subvariety.
Corollary 3.4.3. Let G be a formal group scheme and H be any analytic
group. Let U ⊂ G be a non empty formal open subset of G and ϕ : U → H be
a morphism which respects the group law, i.e. ϕ(u1 + u2 ) = ϕ(u1 ) + ϕ(u2 ) for
u1 , u2 , u1 + u2 ∈ U . Then there is a unique group homomorphism ϕ̂ : G → H
which restricts to ϕ.
Proof. Take any point x ∈ G. We’ll get again an isomorphy between U and
x − U and both are formal open subsets. Since G has an irreducible reduction,
we know that U ∩ (x − U ) is not empty, so we can define ϕ̂(x) as above. We
can again prove locally that ϕ̂ is a morphism and obtain uniqueness since a
morphism is defined by its values on any non empty formal open subset.
The last corollary suggests the definition of a formal rational function as
follows.
41
Definition 3.4.4. A formal rational function on a smooth formal connected
scheme X over R is a pair (f, U ) where U is a non-empty admissible formal
open subset of X and f ∈ OX (U ). Two rational functions (f, U ) and (g, V ) are
called equal if the restrictions of f and g to U ∩ V are equal.
Proposition 3.4.5. Let (f, U ) be a formal rational function on X and let
V ⊃ U be another admissible formal open subset of X. If there is an extension
of f on V , it is unique.
Proof. We can assume that U is a rational subdomain of an affinoid set V . Then
the proposition follows according to the identity theorem of power series.
Remark. On should note that meromorphic functions are not equivalent to
rational functions as one might expect. Just take P1K as U1 = Sp Khζi and
U2 = Sp Khηi glued on the subset Sp Khζ, ηi/(ζη −1). A meromorphic function
without poles can be written as a power series on both U1 and U2 , which
implies that it must be constant. So a meromorphic function on P1K is always
the quotient of two polynomials in ζ as one might expect. But Khζi contains
power series that are not the quotient of two polynomials, for example if |a| > 1
then 1/(ζ − a) has a square root in Khζi as simple calculations show.
Proposition 3.4.6. A short exact sequence of commutative algebraic or formal
analytic groups
ψ
ρ
0→G−
→E−
→B→0
is strict exact if and only if the induced sequence on the tangent spaces
dψ
dρ
0 → TG −→ TE −→ TB → 0
is also exact.
Proof. As in [Ser88, III, §3 Cor. 2 and 3] the condition on the tangent spaces
implies that G is a closed subvariety of E and that B is the geometric quotient
E/G. For any object X in ZC we get the exact sequences
0 → Hom(B, X) → Hom(E, X) → Hom(G, X)
0 → Hom(X, G) → Hom(X, E) → Hom(X, B)
which implies that ρ is the cokernel of ψ and ψ is the kernel of ρ. So the
sequence is strict exact.
Starting with a strict exact sequence we also get
0 → Hom(X, G) → Hom(X, E) → Hom(X, B)
for any X in ZC . But since kernels in ZC and PC are equal, the same sequence
is exact for PC . We can expand C to include non-reduced varieties. Using X =
Spec k[ε] gives the desired sequence on the tangent spaces, with the surjectivity
of the last morphism implied by dimension.
Remark. The condition on the tangent spaces make ψ a reduced closed immersion and ρ a smooth morphism.
42
We know how to describe the extensions of algebraic/analytic groups if the
ψ
ρ
strict exact sequence 0 → G −
→E−
→ B → 0 admits a regular section. We put
2 (B, G) for these extensions.
Hreg
s
If ρ only admits a rational section, we end up with a “birational” group E
with a group law only defined on a non-empty open subset. There is always
a unique algebraic group birationally equivalent to a given birational group as
seen in [Ser88, VII, §1 Prop. 4]. So we still end up with a unique extension this
2 (B, G) for this subgroup of Ext(B, G). Since the extension
way. We put Hrat
s
is constructed by gluing techniques the same works for the formal analytic
case. Note that a rational section of a morphism of formal analytic groups is
a morphism defined on a formal open subset and not necessarily meromorphic.
We will construct this group explicitly for the case that G is a linear group in
Proposition 3.5.3.
Proposition 3.4.7. Let
ρ
0→G→E−
→B→0
be a strict exact sequence of formal groups schemes over R with an algebraically
closed field of fractions with the reduction
e→E
e→B
e→0
0→G
of algebraic groups. Then the sequence has a rational section if and only if the
reduced sequence has a rational section.
Proof. As any section of the analytic sequence reduces to a section of the reduction, this part of the proof is trivial.
On the other hand, since we are talking about rational sections, we can
assume E and B to be affinoid. Then the proof is given by Proposition 1.9.7.
ρ
Proposition 3.4.8. Let 1 → G → E −
→ B → 1 be a strict exact sequence of
connected commutative algebraic groups, where G is linear. Then the sequence
has a rational section.
Proof. We will sketch the proof as it is done in [Ser88, VII, §1 Prop. 6].
Let x be any point of B. Then Ex := ρ−1 (x) = E ×B Spec k(x) is a principal
homogeneous space for G over k(x), i.e. G acts transitively on Ex via multiplication on E and this action is injective for a fixed e ∈ Ex . The action and the
induced division map are both regular.
A rational section of ρ is the same as a k(x) rational point in Ex where x is
the generic point of B, in other words Ex is isomorphic to G and the trivial
homogeneous space.
By a result of Lang and Tate in [LT58, Prop. 4] the isomorphism classes
of principal homogeneous spaces over an arbitrary field k are in one-to-one
correspondence to H 1 (gs , G), where gs is the Galois group of ks /k with the
topology induced by étale extensions, where ks is the separable closure of k
acting on G by action on the coordinates of the points.
43
The group H 1 (gs , G) is trivial for a commutative linear group G. We will only
show this for the case G = Gm,K since this is the case we are most interested in.
The general case can be proven by showing the result for Ga,K and combining
these two cases to form the arbitrary linear group G as seen in [Ser88, VII, §1
Prop. 6].
So for the case G = Gm,K take a finite extension K/k of k with Galois group
A. A cocycle in H 1 (A, Gm,K ) is a map f : A → Gm,K satisfying f (σ ◦ τ ) =
f (σ) · σ(f (τ )) and the coboundry are the maps bα : A → Gm,K σ 7→ σ(α)/α.
Now for a fixed automorphism σ we can calculate the norm of f (σ) in the
extension induced by hσi by the cocycle condition to be
K
NK
hσi (f (σ)) =
r−1
Y
i=0
σ i (f (σ)) =
r−1
Y
i=0
f (σ i+1 )
=1 .
f (σ i )
So according to Hilbert 90 there is an α ∈ K with f (σ) = σ(α)/α i.e. f is a
coboundry.
So since there is only one principal homogeneous space of G over k(x) we
conclude that Ex is isomorphic to G and therefore has a k(x)-rational point
leading to the needed section.
3.5. Extensions by tori
2 (B, G) . A rational section of B to E
There is a different way to interpret Hrat
s
can be translated by the multiplication on B and yields a family of (admissible)
open sets Ui on B, each with a section si . Together with the morphism ψ : G →
E of the exact sequence this gives open embeddings si ◦ p1 × ψ ◦ p2 : Ui × G → E,
making E a fiber space of B with the fiber G. So if GB is the sheaf of germs of
regular functions from B to G, we see that there is a map
2
π : Hrat
(B, G)s → H 1 (B, GB ) .
To make things more explicit let f correspond to the extension E with the
rational section s : U → E where U is an open subset of B. Recall that then
ψ ◦ f (b1 , b2 ) = s(b1 ) + s(b2 ) − s(b1 + b2 )
for any b1 , b2 ∈ U with b1 + b2 ∈ U .
The section s gives an isomorphism ϕ0 between a (formal) open subset of E
and U × G. Covering B with the sets b + U for b ∈ U let us calculate
sb : b + U → B ; b + u 7→ s(b) + s(u)
as the translated sections with ϕb as corresponding isomorphisms.
So for b1 , b2 ∈ U we get for
ϕ−1
b1 ◦ ϕb2 : [(b2 + U ) ∩ (b1 + U )] × G → [(b1 + U ) ∩ (b2 + U )] × G
44
the map defined by
−1
ϕ−1
b1 ◦ ϕb2 (u, g) = ϕb1 (ψ(g) + s(b2 ) + s(u − b2 ))
= ϕ−1
b1 (ψ(g) + s(b1 ) + s(u − b1 )
− s(b1 ) − s(u − b1 ) + s(u) − s(u) + s(b2 ) + s(u − b2 ))
= ϕ−1
b (ψ(g) + sb1 (u) − ψ ◦ f (b1 , u − b1 ) + ψ ◦ f (b2 , u − b2 ))
1
f (u − b2 , b2 )
= u, g ·
f (u − b1 , b1 )
So we see that
π(f ) =
f (u − b2 , b2 )
f (u − b1 , b1 )
bi ∈U
is the image of a cocycle.
Proposition 3.5.1. Let 0 → G → E → B → 0 be a strict exact sequence of
2 (B, G) → H 1 (C, G ) is
formal analytic or algebraic groups. The map π : Hrat
s
B
a group homomorphism which commutes with pullbacks and pushouts.
Proof. As already noted we get π(f ) = (f (u − b2 , b2 )/f (u − b1 , b1 ))bi ∈U as the
image of a cocycle. So we see directly that π(f · g) = π(f ) · π(g) after choosing
a common open set U so π is a group homomorphism.
With that we can directly calculate for any group homomorphism α : G → G0
that
α ◦ f (u − b2 , b2 )
α ◦ f (u − b1 , b1 )
= α ◦ (f (u − b2 , b2 )/f (u − b1 , b1 ))
π(α∗ f )b1 ,b2 =
= α∗ π(f )b1 ,b2
holds and that for any group homomorphism γ : B 0 → B
f (γ(u − b2 ), γ(b2 ))
f (γ(u − b1 ), γ(b1 ))
f (γ(u) − γ(b2 ), γ(b2 ))
=
f (γ(u) − γ(b1 ), γ(b1 ))
= γ ∗ π(f )γ(b1 ),γ(b2 )
π(γ ∗ f )γ(b1 ),γ(b2 ) =
holds as well, which concludes the proof.
Proposition 3.5.2. Let G be a commutative linear and B be a commutative
2 (B, G) →
projective algebraic or formal analytic group. Then the map π : Hrat
s
1
H (B, GB ) is injective.
2 (B, G) and
Proof. Let again be E the extension corresponding to f ∈ Hrat
s
s : U → E the rational section of ρ, defined on the open subset U of B. If f
is in ker π then f (x − b, b) = s(x − b) + s(b) − s(x) is regular on all of U × U
after the change by a coboundry. But then s is a regular section which in turn
implies that f is regular on B × B and since B is projective and G is affine, f
is necessarily constant and thus induced by the constant coboundry.
45
Proposition 3.5.3. Under the assumptions of Proposition 3.5.2 the image of
2 (B, G) → H 1 (B, G ) consists of all elements x ∈ H 1 (B, G )
the map π : Hrat
s
B
B
∗
for which mB x = p∗1 x · p∗2 x holds on B × B.
2 (B, G) we get
Proof. First we note that for any f ∈ Hrat
s
m∗B π(f ) = π(m∗B f ) = π(p∗1 f · p∗2 f ) = p∗1 π(f ) · p∗2 π(f )
according to Lemma 3.5.1 and the coboundaries of H 2 (B × B, G).
On the other hand, if (Ui )i∈I is an open covering for which (gij ) represents
an element in H 1 (B, GB ) we can reformulate the stated property, as
gij (x + y)
fj (x, y)
=
gij (x)gij (y)
fi (x, y)
where (fi ) is in Z 0 (B × B, GB×B ). We want to show that any fi represents a
preimage of (gij ).
2 (B, G) , all of the (f ) differ only by a coboundry.
By the definition of Hrat
s
i
For any i the fi also fulfills the cocycle condition, since
h(x, y, z) :=
fi (y, z)fi (x, y + z)
fi (x + y, z)fi (x, y)
defines the same function for all i ∈ I and is thus constant as a regular function
from the projective variety B × B × B to the affine variety G and h(0, 0, 0) = 1.
This map is obviously a group homomorphism. It is also injective. Suppose
we have (gij ) in the kernel i.e.
gij (x + y)
hj (x + y)hi (x)hi (y)
=
.
gij (x)gij (y)
hi (x + y)hj (x)hj (y)
Changing (gij ) by the coboundry (hj /hi ) gives
gij (x + y) = gij (x) · gij (y) .
So gij is a group homomorphism from B to G by Proposition 3.4.2. Therefore
gij is constant as a map from a projective to an affine space and (gij ) is the
trivial bundle.
2 (B, G) this trivialization for π(f ) is given by
For f ∈ Hrat
s
π(f )bi ,bj (x + y)
f (x + y − bj , bj ) · f (x − bi , bi ) · f (y − bi , bi )
=
π(f )bi ,bj (x) · π(f )bi ,bj (y)
f (x + y − bi , bi ) · f (x − bj , bj ) · f (y − bj , bj )
f (x, y) · f (x − bi , y) · f (y − bi , bi )
=
f (x − bj , y) · f (y − bj , bj ) · f (x, y)
f (x − bi , y − bi ) · f (x + y − 2bi , bi )
=
f (x − bj , y − bj ) · f (x + y − 2bj , bj )
which for bi = 0 recovers the cocycle f .
46
Remark. One can even construct an explicit extension of B by G this way. Take
a principal fiber space L of x. Then m∗B x = p∗1 x + p∗2 x gives a regular function
g : L × L → L compatible with the multiplication on B. One can use this
function g to define a group law on L, see [Ser88, pg. 182 et seq.] for details.
Since the construction works as well for analytic groups, this shows that all
rational cocycles indeed come from an extension.
Proposition 3.5.4. Let B be an abelian variety or the generic fiber of a proper
formal group scheme. Then an extension of B by Gm,K is equal to a point of
Picτ (B), the dual abelian variety of B.
Proof. We have already seen in Proposition 3.5.2 that the morphism
×
2
π : Hrat
(B, Gm,K )s → H 1 (B, OB
)
×
is injective and that its image consists of the line bundles L ∈ H 1 (B, OB
) which
∗
∗
∗
fulfill mB L = p1 L ⊗ p2 L which by the Theorem of the square as in [Mum08, II.
8./III. 13.] happens if and only if L ∈ Picτ (B).
Proposition 3.5.5. Let B be an abelian variety or the generic fiber of a proper
formal group scheme and t ∈ N. Then an extension of B by Gtm,K is equal to a
point of Picτ (B)t .
Proof. This follows from the previous proposition and the fact that the coho×t
× t
mology group H 1 (B, OB
) equals H 1 (B, OB
) by definition.
We can formulate the last proposition without introducing coordinates.
Corollary 3.5.6. Let B be a proper formal group scheme. Then an extension of the generic fiber of B by a torus T is equal to a group homomorphism
Hom(T, Gm,K ) → Picτ (B) of the character group of T to the translation invariant line bundles over B.
Proof. We can choose coordinates on T so that we get T →
˜ Gtm,K . An extension
t
1 → Gm,K → E → B → 1 gives rise to the map
ϕ : Hom(Gtm,K , Gm,K ) → Picτ (B)
via α 7→ α∗ E which is a total fiber space of a line bundle in Picτ (B). By the
definition of the pushout, this is a group homomorphism. On the other hand
any such group homomorphism is determined by the images of the projections
which give the point of Picτ (B)t of Proposition 3.5.5.
Corollary 3.5.7. Let 0 → T → E → B → 0 be an extension of the generic fiber
of a proper formal group scheme by a torus T . An S-valued point σ : S → E
is equivalent to a family of S valued points σχ : S → PB×φ(χ) with σχ1 +χ2 =
σχ1 ⊗ σχ2 and where φ is the homomorphism of the last corollary and PB×B 0 is
the Poincaré bundle over B.
Q
In other words we can write E = ni=1 PB×φ(e0i ) where e0i is a basis of the
character group and a point t = (t1 , . . . , tn ) ∈ E corresponds to the family
⊗m0
⊗m0
tχ = t1 1 ⊗ · · · ⊗ tn n ∈ PB×φ(χ) where χ = m01 e01 + · · · + m0n e0n with m0i ∈ Z.
47
Note. If one does not care about the embedding ψ : Gm,K → E of Gm,K in
the extension E, one needs to divide by the automorphism group of Gm,K .
Therefore two line bundles L and L−1 yield the same extension up to ψ.
Definition 3.5.8. Let 1 → Ḡtm,K → Ē → B → 1 be an extension of formal
group schemes. Denote by 1 → Gtm,K → E → B the pushout given by the map
t
Ḡtm,K → Gtm,K on the generic fibers. We define |·| : E → R+
0 by x 7→ |a| with
t
−1
a ∈ Gm,K and x · a ∈ Ē. A lattice M in E is an analytic subgroup of E so
that |·||M is injective and − log|M | is a lattice in Rt . The rank of M is just the
rank of − log|M |.
Remark. There is a more formal approach to define a value on E. We know
that E is a line bundle over the formal scheme B. So the defining cocylces (gij )
have absolute value 1 as they are functions of a formal scheme to Gm,K . This
induces the absolute value on E with the absolute value of Gm,K and leads to
the same value that we defined.
Proposition 3.5.9. Let E be an extension of the generic fiber of a proper
formal group scheme B by a Torus T as before. Let M be a lattice of full rank
in E. Then E/M is again a proper analytic group variety.
Proof. We write
M
0
/T
ψ
/E
ρ
/B
/0
A
with A = E/M . As an abstract group A obviously has a group structure. E
is a total fiber space of a line bundle, so we can find an affinoid subset U ⊂ B
such that E can be covered by a finite amount of translations of ρ−1 (U ) and
ρ−1 (U ) = Gtm,K × U . We write Tc for the annulus T hc−1 ζ, cζ −1 i where |c| < 1
and ζ is a coordinate on T . Since the rank of the lattice is full we find c ∈ K ×
such that the translates of Tc × U already cover A. Since B is assumed to be
proper and is formal so its generic fiber is proper in the rigid analytic sense and
we can cover U by relatively compact subsets. This means that A is proper.
Proposition 3.5.10. Let A be proper analytic group and let Ē be an open
formal analytic subgroup of A. Suppose there is a morphism ψ : Gtm,K → A
that restricts to ψ̄ : Ḡtm,K → Ē and gives Ē as extension of a proper formal
group scheme B by Ḡtm,K . Suppose that there is an ε ∈ R such that every
element of A can be written as ψ(g) · e with e ∈ Ē and ε ≤ |g| ≤ ε−1 . Then A
is isomorphic to E/M where E is the pushout as in Definition 3.5.8 and M is
a lattice of full rank.
Proof. We can write the elements of E as tuples (g, e) ∈ Gtm,K × Ē with the
equivalence relation (g, e) ∼ (g 0 , e0 ) if and only if ψ̄(g/g 0 ) = e0 /e according to
Proposition 3.3.4.
48
Define the group homomorphism η : E → A by setting η(g, e) = ψ(g) · ι(e)
with ι : Ē → A as the open embedding of Ē in A. The kernel of η is a lattice
in E, since ι and ψ̄ are injective and the value of an element (g, e) of E is the
value of g.
The assumption that every element of A can be written as ψ(g) · e where
g is bounded and e ∈ Ē implies the surjectivity and that the lattice is of full
rank.
49
4. The Jacobian of a formal analytic
curve
4.1. The cohomology of graphs
In this section G(V, E) will be an arbitrary connected graph with vertex set V
and edge set E.
Definition 4.1.1. A graph G(V 0 , E 0 ) with V 0 ⊂ V and E 0 ⊂ E, where every
edge of E 0 has source or target in V 0 is called a subgraph of G(V, E). A subgraph
is called complete, if E 0 contains all edges of E with source and target in V 0 .
We want to discuss homology and cohomology of a connected graph. For this
to be defined, we need to fix a topology on G. This is done by declaring all
complete subgraphs of G to be open. Since cycles in a homology group have an
orientation, we will have to choose an arbitrary orientation for each edge e ∈ E.
In this section a simple cycle is a directed closed path in the graph which
visits an edge only once. The general term cycle is used in the homological sense
of the word, i.e. a chain of edges with zero boundary. Cycles can be obtained
by taking formal sums of simple cycles modulo the standard concatenation of
cycles. We say that an edge ends in a vertex if said vertex is either target or
source of the edge. The graph G may contain loops.
Definition 4.1.2. Denote by n the number vertices of G and by e the number
of edges. The number t := e − n + 1 is called the cyclomatic number of G.
The cyclomatic number of a graph is fundamental to describe its homology,
as the following proposition shows.
Proposition 4.1.3. Let t be the cyclomatic number of a connected graph G.
Then one can fix t edges ε1 , . . . , εt in E such that G0 (V, E \ {ε1 , . . . , εt }) is a
tree. Each of these edges εi corresponds uniquely to a simple cycle γi on G in
a natural way.
Proof. A connected graph is a tree if and only if its cyclomatic number t equals
0. If a connected graph G contains cycles, i.e. t ≥ 1 we remove an edge ε1
of one of the cycles, thereby getting a graph Ĝ with cyclomatic number t − 1.
Repeating that t times yields a tree and t edges ε1 , . . . , εt . The graph Ḡ(V, E \
{ε1 , . . . , εi−1 , εi+1 , . . . , εt }) has exactly one simple cycle matching orientation
with εi which we denote by γi . In other words this is the unique cycle in G
which passes exactly one time through εi but none of the εj for j 6= i.
To compute the homology groups of G we first note that the homology groups
for the constant sheaf Z defined by our topology coincide with the homology
51
groups of G if we regard G as a simplicial complex of dimension 1. This gives
us the group H1 (G, Z) as the group of formal sums of the closed paths of G
with coefficients in Z, i.e. the cycles of G.
Proposition 4.1.4. The group H1 (G, Z) is isomorphic to Zt with t the cyclomatic number of Definition 4.1.2. Higher homology groups vanish.
Proof. We define a group homomorphism from cycles in G to Zt , by counting
how often the cycle passes through the edges ε1 , . . . , εt of Proposition 4.1.3,
where the sign encodes the direction of the cyclePin this edge. For an element
(ai )ti=1 of Zt we obtain a preimage as the cycle ti=1 ai γi . Suppose two cycles
γ and γ 0 have the same image. Then γ − γ 0 is mapped to 0, i.e. it is equivalent
to a cycle which passes through none of the εi . But G0 (V, E \ {ε1 , . . . , εt }) is a
tree, thus γ − γ 0 is trivial and γ = γ 0 .
We can now compute the cohomology groups H q (G, Z) via Čech cohomology.
Proposition 4.1.5. Let (A, ·) be any abelian group. Then the cohomology group
H 0 (G, A) is isomorphic to A. The group H 1 (G, A) arises as Hom(H1 (G, Z), A)
and is as such isomorphic to At . All higher cohomology groups vanish.
Proof. A graph is connected in the graph-theoretical sense if and only if it is
connected topologically. Therefore H 0 (G, A), the group of global section of the
constant sheaf A, is trivially isomorphic to A.
To calculate the higher cohomology groups we define an open covering. Let
Ge be the subgraph of G consisting of the edge e and the two vertices it joins.
Then G = {Ge ; e ∈ E(G)} is an open covering of G, as the Ge are complete.
With respect to this covering, we have
Y
Č q (G, A) =
A(Ge0 ∩ · · · ∩ Geq )
(e0 ,...,eq )∈E(G)q+1
and
dq : Č q → Č q+1 ; α 7→
p+1
Y
!
(−1)k αi0 ,...,ik−1 ,ik+1 ,...,iq+1
k=0
,
i0 ,...,ip+1
as in [Har77, pg. 218] for standard Čech cohomology. The elements of Č 0 (G, A)
therefore assign an element of A to each edge of E, whereas the elements of
Č 1 (G, A) assign an element of A to each pair of edges which share a vertex of
G.
To get the group Ȟ 1 (G, A) = ker d1 / im d0 we need to discuss the kernel of d1
and the image of d0 . As always, the elements of ker d1 are the elements of Č 1
which satisfy the cocycle relations αei ,ej = αei ,ek · αek ,ej for every three edges
ei , ej , ek sharing a vertex v. In particular, we have αe,e = 1 and αei ,ej = αe−1
.
j ,ei
1
To finish our description of Ȟ (G, A), we need to look at the coboundaries
im d0 . By definition, these are the elements defined by αei ,ej = βej βe−1
for a
i
0-cocycle (βe )e∈E .
52
4.1. The cohomology of graphs
Given any cycle γ =
Pm
k=1 ek
m
Y
∈ H1 (G, Z), we get
αek ,ek+1 =
k=1
m
Y
=1 ,
βek+1 βe−1
k
(4.1)
k=1
where e1 = em+1 .
Conversely, suppose we have an element α of Č 1 (G, A) such that the product
of (4.1) is trivial for every cycle of G. We can define a 0-cocylce (βe )e∈E of G by
, if βej is already defined
setting βe0 = 1 and recursively defining βei = βej αe−1
i ,ej
and ei and ej share a vertex. Since the product of (4.1) is trivial this does not
depend on the path we choose from ei to e0 .
This means that the elements of Ȟ 1 (G, A) are uniquely defined by the possible
values of the left hand side of (4.1). Therefore we get a dualizing form
Φ : H1 (G, Z) × Ȟ 1 (G, A) → A
by sending (γ, α) to this left hand side. This gives us an isomorphism
ϕ : Ȟ 1 (G, A) →
˜ At
by dualizing using the generators (γi ) of H1 (G, Z).
By the same argument the higher cohomology groups must be dual to the
corresponding homology groups. Since these vanish, the cohomology groups
vanish too.
Since any intersection of elements of G is connected or empty the sheaf A is
acyclic on each of these intersections. This implies that H q (G, A) = Ȟ q (G, A)
for all q ∈ N.
Definition 4.1.6. For an arbitrary edge e with target vertex v and an element
a ∈ A we define the weighted cocycle α(e, a) = (αei ,ej )(ei ,ej )∈E 2 by setting
αei ,ej


a
= a−1


1
if ei = e, ej 6= e and ej ends in v
if ej = e, ei 6= e and ei ends in v
otherwise.
One checks easily that every weighted cocycle obeys the cocycle conditions.
Recall that an edge ends in a vertex if said vertex is either target or source.
Lemma 4.1.7. The image ϕ(α(e, a)) of a weighted cocycle under the map
ϕ : H 1 (G, A) → At can be computed as


a
ϕ(α(e, a))i = a−1


1
if e ∈ γi and γi traverses e in the same direction
if e ∈ γi and γi traverses e in the other direction
otherwise.
where γi are the simple cycles of Proposition 4.1.4.
53
Proof.PIf the cycle γi does not contain e, every factor of (4.1) is trivial. For
γi = m
k=1 ek with er = e and α(e, a) = (αej ,ek )(ej ,ek )∈E 2 , we have
ϕ(α(e, a))i =
m
Y
αek ,ek+1 = αer−1 ,er αer ,er+1 .
k=1
If γi traverses e in the direction of e we get αer−1 ,er = 1 and αer ,er+1 = a. If on
the other hand γi traverses e in the opposite direction we have αer−1 ,er = a−1
and αer ,er+1 = 1.
Corollary 4.1.8. Denote by ê the edge e with reversed direction. Then we get
α(ê, a) = α(e, a−1 ).
Since the cycle γi corresponding to the edge εi was defined as the cycle
containing εi and none of the εj for i 6= j, we see that
ϕ(α(εi , a)) = (1, . . . , 1, a, 1, . . . , 1) ,
the element of At with a at the ith component. This means that the weighted
cocylces are dual to the simple cycles in the sense that
Φ(α(εi , a), γj ) = aδij .
With this construction we can give an explicit inverse morphism of ϕ. Let
(ai )ti=1 be an element of At . Then the cocycle
(αej ,ek )(ej ,ek )∈E 2 =
t
Y
α(εi , ai )
i=1
is a preimage of (ai ).
4.2. The cohomology of curves with semi-stable
reduction
We recall the definition of a semi-stable curve of Chapter 2.
Definition 4.2.1. A projective, connected, geometrically reduced curve X̃/k
that has only ordinary double points as singularities is called semi-stable.
Let X be a formal analytic curve with semi-stable reduction X̃. We write
X̃i , i = 1, . . . , n for the irreducible components of the reduction. The components X̃i can at most have nodes as singularities. If there is a node which lies
only on one component, we know by Chapter 2 that it has a formal fiber which
is isomorphic to an open annulus. We can use a formal blow-up to subdivide
this open annulus in order to replace this node by two nodes lying on different
components. Since all statements in this section are trivial for a curve with good
reduction, this means that we can assume that the reduction X̃ has at least two
irreducible components and that every node lies on two different components.
54
4.2. The cohomology of curves with semi-stable reduction
Example 4.2.2. To get an example for a curve with semi-stable reduction and
non-regular components, we look at the Weierstraß-equation η 2 ζ − ξ(ξ − ζ)(ξ −
λζ) defining an elliptic curve X in P2K for a λ ∈ K with |λ| < 1 and coordinates
(ξ, η, ζ). The standard affinoid cover of P2K with three copies of B2K gives the
nodal curve defined by T02 T2 − T12 (T1 − T2 ) in P2k as the reduction of X. Blowing
up (ξ, λ) in this model and changing coordinates in a suitable manner gives us
the curve of Example 1.11.4. The reduction of X̃ now has two components
X̃1 , X̃2 of genus 0, which are regular.
Definition 4.2.3. For a semi-stable curve X̃, we want to define the dual graph
G(X̃). The vertex set V of G is the set of irreducible components of X̃. The
edge set E of G is the set of double points of X̃, where an edge e connects the
vertices corresponding to the components which intersect in the corresponding
double point.
For a semi-stable curve X̃ we will denote the double points corresponding to
the edges εi (as in Proposition 4.1.3) by ỹi .
Example 4.2.4. The dual graph G(X̃) of the elliptic curve of Example 1.11.4
has two vertices and two edges between these edges. Removing one of these
edges yields a tree. We denote the double point corresponding to this edge as
ỹ.
We now apply the results for graphs of Section 4.1 to a p-adic curve X with
semi-stable reduction X̃.
Proposition 4.2.5. Let X be a smooth projective formal analytic curve with
semi-stable reduction X̃. Let G(V, E) be the dual graph of X̃. For any abelian
group A we have
H q (X, A) = H q (X̃, A) = H q (G, A) .
Proof. Let e ∈ E be an edge corresponding to a double point x̃ between the
components X̃i and X̃j . Define
Ze := π −1
X̃i ∪ X̃j \ Sing X̃ ∪ x̃ .
By our assumption, x̃ is not the only singularity of X̃, so Z := {Ze ; e ∈ E}
is an open affinoid covering of X. The part of the graph associated to the
reductions Z̃e are the Ge of the proof of Proposition 4.1.5. This implies that
Ȟ q (Z, A) = Ȟ q (Z̃, A) = Ȟ q (G, A).
To prove that Ȟ q (Z, A) = H q (X, A) it suffices to show that A is acyclic on
the intersections of elements of Z. This can be done by a case-by-case analysis
of the intersections as in [BL84, Prop. 2.2].
Corollary 4.2.6. The isomorphism : H 1 (G, A) →
˜ At gives rise to the isot
1
×
t
1
morphism ψ : Gm,K →
˜ H (X, K ), ψ̄ : Ḡm,K →
˜ H (X, R× ) and its reduction
ψ̃ : Gtm,k →
˜ H 1 (X̃, k × ). This gives these cohomology groups are canonical structure as analytic or algebraic groups.
55
4.3. The Jacobian of a semi-stable curve
Let X̃ be a semi-stable curve. As seen in Chapter 3 the Picard group Pic X̃ of
X̃ is defined as the isomorphism classes of line bundles over X̃. This group is
×
canonically isomorphic to H 1 (X̃, OX̃
). In this situation the group CaCl(X̃) of
Cartier divisors modulo linear equivalence coincides with Pic X̃.
Proposition 4.3.1. The group Pic X̃ can be represented as Div X̃/ ∼ where
Div X̃ are the Weil divisors on X̃ with support in the non-singular locus of X̃
and ∼ is the usual linear equivalence.
Proof. See [BL84, pg. 274]
Note that our definition of the Jacobian variety in Chapter 3 is only valid for
non-singular curves. We can now generalize this concept.
The Weil divisors with support in the non-singular locus have a well-defined
degree on each irreducible component of X̃.
Definition 4.3.2. The Jacobian variety Jac(X̃) of X̃ is the subgroup of Pic(X̃)
composed of the classes of divisors with degree 0 on each component.
×
Theorem 4.3.3. The natural map H 1 (X̃, k × ) → H 1 (X̃, OX̃
) induces a short
strict exact sequence
ψ̃
ρ̃
1 → Gtm,k −
→ Jac X̃ −
→ B̃ :=
Y
Jac X̃i → 1
(4.2)
Proof. The second morphism ρ̃ is obtained by projecting a divisor pointwise
onto the irreducible components of X̃, i.e. the pull-back fi∗ L of a line bundle L
by the canonical immersion fi : X̃i → X̃. This map is obviously surjective and
compatible modulo principle divisors.
Let now [D] ∈ ker ρ̃. This means that [D|X̃i ] = 1, i.e. D|X̃i = (fi ) for a
rational function fi on X̃i . Since D has support in the non-singular locus, every
fi is defined and non-zero in every double point x̃ of X̃. Let x̃1 , x̃2 ∈ X̃i be two
double points on the same component of X̃ with corresponding edges e1 , e2 with
a common vertex v in the dual graph of X̃. Setting αe1 ,e2 = fi (x̃1 )/fi (x̃2 ) yields
an element of H 1 (X̃, k × ) which corresponds to the natural map H 1 (X̃, k × ) →
×
H 1 (X̃, OX̃
).
On the other hand take an element α ∈ H 1 (X̃, k × ). We know by a well-known
corollary of the Riemann–Roch Theorem that there exists a function with given
values in the finitely many double points of X̃i , so we can set the values at the
given double points as we wish. Therefore every element of H 1 (X̃, k × ) yields
appropriate functions as before. Since every component of X̃ is a projective
curve, the divisor corresponding to these function will have degree 0 on each
component. This proves the exactness of the sequence.
To see that (4.2) is strict exact, we have to look at the geometric structure of
Jac X̃. Without restriction, we can assume that every marked double point x̃i
is blown up to a component isomorphic to P1k , intersecting the rest of X̃ in the
points at zero and infinity. A collection (f˜i ) of functions can be renormed at
56
the unmarked double points and represented by (ci Z − 1)/(Z − 1) for a suitable
ci ∈ Gm,k on the blown up P1k with coordinate Z for the marked double point
x̃i , yielding a rational function on all of X̃.
This way, we can set
X̃ (∗)0 = Gtm,k ×
n
Y
(gi )
X̃i
i=1
(g )
where X̃i i is the gi th symmetric product of the ith component with gi as
the genus of that component. Thereby we can construct the Jacobian
Qvariety
with the usual Weil construction, see for example [Mil86, §7]. Since Jac X̃i
is constructed the same way, we see that ψ̃ embeds Gtm,k as a closed subvariety
and that ρ̃ is smooth.
We can now directly describe the group law of the Jacobian by the methods
of Chapter 3. We need to make the Weil construction of the Jacobian more
explicit. We select a non-singular base point x̃i for every component X̃i of X̃.
(g )
Then Jac X̃i has an open subset Ui isomorphic to an open subset of X̃i i where
gi is the genus of X̃i given via the morphism
(gi )
X̃i
→ Jac X̃i ; D 7→ [D − gi x̃i ]
We can without restriction assume that Ui has no support in the singular points
of X̃. On Ui , the group law is given via D1 + D2 = div hi + D1 + D2 − gx̃i
where hi is the global section of L(D1 + D2 − gx̃i ), unique up to a constant.
Q (g )
Let X̃ (∗) = X̃i i and let U be the product of the
P subsets Ui . On U we can
define a section of ρ by sending D ∈ X̃ (∗) to [D − gi x̃i ] on Jac X̃. This map
defines the open subset Gtm,k × U of Jac X̃ with the group law
(a1 , D1 ) + (a2 , D2 ) = (a1 a2 f (D1 , D2 ), D1 + D2 ) .
The cocylce f is defined by selecting the hi so that they agree on the unmarked
double points and then taking the quotient of the two values at the double point
corresponding to εi as the ith component as described in the proof of Theorem
4.3.3.
Proposition 4.3.4. Let X̃ 0 be a regular curve of genus g and x̃1 , x̃2 two points
on X̃ 0 . Denote by X̃ the semi-stable curve obtained by identifying x̃1 and x̃2 on
X̃ 0 . If x̃1 = x̃2 this gives a rational nodal curve glued with X̃ 0 in x̃1 . The line
bundle of Proposition 3.5.4 corresponding to the extension (4.2) is given by the
point [x̃2 − x̃1 ] of Jac X̃ 0 regarded as a line bundle on Jac X̃ 0 via the autoduality
Jac X̃ 0 → Pic0 (Jac X̃ 0 ) given by any base point x̃0 .
Proof. Proposition 3.5.4 states that (4.2) means that Ẽ := Jac X̃ is a Gm,k torsor associated to a line bundle over B̃ := Jac X̃ 0 .
Recall that this line bundle is given by a rational section s : Ũ → E of ρ in
ψ
ρ
0 → Gm,k −
→ Ẽ −
→ B̃ → 0
57
together with its translations sc̃ : c̃ + Ũ → Ẽ as sc̃ (x̃) = s(x̃ − c̃) + s(c̃) for any
c̃ ∈ Ũ . These give isomorphisms
ϕc̃ : Gm,k × (c̃ + Ũ ) → ρ−1 (c̃ + Ũ )
by ϕc̃ (ã, ũ) = ψ(ã) + sc̃ (ũ).
A rational section t of ρ gives rise to a rational function tc̃ = p1 ◦ ϕ−1
c̃ ◦ t for
tc̃ : c̃ + Ũ → P1 . By the definition of ϕc̃ we get tc̃ = p1 ◦ (t − sc̃ ). Since poles
and zeroes of tc̃ behave well when changing c̃ this gives the Cartier divisor on
C that belongs to the extension.
So to calculate the divisor associated to Jac X as a line bundle we only need
to determine the divisor of s.
With [Mil86, Lemma 6.9./ Remark 6.10.] we know that
ι∗ : Picτ (Jac X̃ 0 ) → Pic0 X̃ 0
is an isomorphism, where
ι : X̃ 0 → Jac X̃ 0 ; x̃ 7→ [x̃ − x̃0 ]
is the canonical embedding with respect to x̃0 .
So when we fix a point x̃0 on X̃ 0 any divisor class on Jac X̃ 0 is defined by
an element of Jac X̃ 0 . This leaves us with calculating the divisor of s ◦ ι as
described above.
Let Ũ ⊂ (X̃ 0 \ {x̃1 , x̃2 })(g) be the set of effective divisors with degree g such
that D − gx̃0 is non special for all D ∈ U . Since divisors of the form x − x0 are
always non special we see that ι|X̃ 0 \{x̃1 ,x̃2 } is restricted to Ũ .
We get a rational section s : Ũ → Jac X̃ of (4.2) by sending [D − gx̃0 ] ∈
Jac X̃ 0 = B̃ to [D − gx̃0 ] ∈ Jac X̃ = Ẽ. Choose any Dc̃ ∈ Ũ such that Dc̃ 6= gx̃0
and that Di ∈ Ũ where Di is defined by
[x̃i − x̃0 ] + [Dc̃ − gx̃0 ] = [Di − gx̃0 ] ,
for i = 1, 2.
If D is any effective divisor in Ũ ∩ Dc̃ + Ũ we can define D0 as the unique
effective divisor that fulfills
[D − gx̃0 ]Jac X̃ 0 − [Dc̃ − gx̃0 ]Jac X̃ 0 = [D0 − gx̃0 ]Jac X̃ 0
and we get
sc̃ ([D − gx̃0 ]Jac X̃ 0 ) = p1 ◦ ϕ−1
c̃ (s([D − gx̃0 ]Jac X̃ 0 ]))
= p1 s [D − gx̃0 ]Jac X̃ 0 − sc̃ [D − gx̃0 ]Jac X̃ 0
= p1 s [D − gx̃0 ]Jac X̃ 0
−s [D − gx̃0 ]Jac X̃ 0 − [Dc̃ − gx̃0 ]Jac X̃ 0
−s [Dc̃ − gx̃0 ]Jac X̃ 0
.
58
(4.3)
We can now use 4.3 to simplify the second summand to get
= p1 s([D − gx̃0 ]Jac X̃ 0 ) − s([D0 − gx̃0 ]Jac X̃ 0 )
−s([Dc̃ − gx̃0 ]Jac X̃ 0 )
= p1 ([D − gx̃0 ]Jac X̃ − [D0 − gx̃0 ]Jac X̃ − [Dc̃ − gx̃0 ]Jac X̃ )
= hD (x̃1 )
where hD ∈ M0X is a rational function with the divisor
div hD = D0 + Dc − D − gx̃0
and hD (x̃2 ) = 1 by the definition of s and ψ.
We can interpret the hD as a rational function h on Jac X 0 × X 0 . To find
the divisor associated to the extension, we need to find the zeroes and poles of
h(·, x̃1 ). By applying ι we can interpret h as a rational function on X 0 × X 0 and
reduce the problem to finding the zeroes and poles of the first argument.
We can choose Dc in such a way that the support of Dc̃ , D1 and D2 does
not contain x̃1 and x̃2 . Using the definition of h on X 0 × X 0 we see that the
diagonal is a simple pole. This is the only component of the divisor that meets
the point (x̃1 , x̃1 ) by our assumption on Dc̃ .
Therefore h(ι(x̃), x̃1 ) has a single pole at x̃1 and by the same argument a
single zero at x̃2 .
This means that Jac X is a total fiber space of the divisor class [x̃2 − x̃1 ] over
Jac X̃ 0 for any base point x̃0 ∈ X̃ 0 .
If x̃1 = x̃2 , we see that the rational functions on X are the same as on X 0 ,
thus the extension corresponds to the trivial line bundle.
Proposition 4.3.5. Let X̃ be a semi-stable curve. Denote by x̃i,γ,1 and x̃i,γ,2
the two double points of the component X̃i which lie on the cycle γ with the
corresponding orientation.
The line bundles given by the extension (4.2) are
N
given by Lγ = [x̃i,γ,2 − x̃i,γ,1 ].
Proof. We can use the same proof as in 4.3.4 by replacing X̃ 0 with the disjoint
union of the irreducible components X̃i of X̃. We fix a base point x̃i for every
irreducible
component and
N
Nlet s be the section induced by these base points, i.e.
s( [Di − gi x̃i ]Jac X̃ 0 ) = [Di − gi x̃i ]Jac X̃ for suitable effective divisors Di on
X̃i . We can choose a divisor Di,c̃ with support solely in X̃i to translate s and
thereby calculate the divisor of s on the component X̃i by the same method as
in 4.3.4 which gives Lγ |X̃i = [x̃i,γ,2 − x̃i,γ,1 ]. This proves the assumption.
Proposition 4.3.6. The sequence (4.2) of Theorem 4.3.3 will split if and only
if the vertices belonging to the simple curves γi are rational curves.
Proof. By the virtue of Proposition 4.3.5, we only need to determine when the
bundle [x̃2 − x̃1 ] on every component X̃i is trivial. If x̃1 6= x̃2 the triviality of
the bundle gives an isomorphism of X̃i to P1 , so X̃i is rational.
59
4.4. The Jacobian of a curve with semi-stable reduction
To apply this result to the original curve X, we lift the exact sequence of
Theorem 4.3.3.
We want to describe the set of Weil divisors analogous to the Weil divisors
used in Definition 4.3.2.
Definition 4.4.1. We say that a Weil Divisor D on X has support in the
non-singular locus of the reduction if we have supp D ⊂ π −1 (X̃ \ Sing X̃).
We say that D has degree 0 in the reduction if D̃ has degree 0 on every
irreducible component of X̃.
The Divisors with support in the non-singular locus of the reduction and
degree 0 in the reduction are denoted by DivX
With [BL84, Thm. 5.1] we know that DivX/ ∼ yields an open analytic
subgroup J¯ of Jac X with the canonical reduction Jac X̃.
Theorem 4.4.2. There is a strict exact sequence
ψ̄
ρ̄
1 → Ḡtm,K −
→ J¯ −
→B→1
(4.4)
which reduces to (4.2) of Theorem 4.3.3.
Proof. See [BL84, Theorem 6.6].
Note. If the sequence (4.4) splits, then the exact sequence (4.2) of the reduction
will also split, since the splitting morphism can be reduced. On the other hand
knowing that the sequence of the reduction splits is not sufficient to deduce
whether or not sequence (4.4) splits.
We want to extend this exact sequence to the whole Gtm,K . This can be done
via the theory of group objects as we have seen in the previous chapter.
Theorem 4.4.3. There is a exact sequence
ψˆ
J
1 → Gtm,K −−→
Jˆ → B → 1
(4.5)
with
Jˆ := Gtm,K × J¯ / (g, j) ∈ Ḡtm,K × J¯ ; ψ̄(g) = j −1 .
Proof. There is a unique group homomorphism ϕ : Ḡm,K → Gm,K , so Jˆ is just
¯
the push forward ϕ∗ J.
The projection on the first factor of the direct product gives a surjective
morphism
ϕ : Jˆ → |K × |t → Rt ; (g, j) 7→ |g| 7→ − log|g| .
(4.6)
In the rest of this section we will discuss how we can use the analytic group
ˆ
J to describe the Jacobian Jac X.
Definition 4.4.4. Let O̊X be the sheaf of functions with supremum norm 1 on
×
every component. Let O̊X
be the subsheaf of functions without zeros.
60
Definition 4.4.5. With the natural morphisms, we can regard H 1 (X, K × ) and
×
×
H 1 (X, O̊X
) as part of H 1 (X, OX
). We call the bundles coming from H 1 (X, K × )
×
toric and the bundles coming from H 1 (X, O̊X
) formal. Note that the bundles
in H 1 (X, R× ) are both toric and formal.
×
Lemma 4.4.6. Every element f ∈ H 1 (X, OX
) can be written as a product
×
1
×
f = α·g of a toric bundle α ∈ H (X, K ) with a formal bundle g ∈ H 1 (X, O̊X
).
×
Proof. Let f = (fi,j ) ∈ H 1 (X, OX
) be represented by a cocycle on an formal
open covering U = (Ui )i∈I which is a refinement of the covering Z. With the
appropriate
blow-up
and further refinement we can achieve that Ui ∩ Uj ⊂
π −1 X̃k \ Sing X̃ for every i 6= j. Blowing up double points of the reduction only adds curves of genus 0, on which every divisor is trivial. Hence the
×
cohomology group H 1 (X, O̊X
) remains unchanged.
We know thatthesupremum norm is multiplicative on intersections Ui ∩
Uj ⊂ Xk := π −1 X̃k and thus that the absolute value of fi,j is well defined.
We can find a cocycle α = (αi,j ) ∈ Č 1 (U, K × ) with |αi,j | = |fi,j |. Setting
gi,j = fi,j /αi,j gives us a cocycle g with absolute value 1 on each component,
×
).
i.e. g ∈ Č 1 (U, O̊X
We now calculate these factorizations explicitly for line bundles corresponding
to divisors of the form [x1 − x2 ], x1 , x2 ∈ X. For a line bundle L denote by DL
×
) the representing element of
the corresponding divisor and by fL ∈ H 1 (X, OX
the cohomology group.
Lemma 4.4.7. An analytic Weil divisor with support in the non-singular locus
of the reduction can be solved locally and thus be written as a Cartier divisor.
Proof. Let x ∈ X be a point with the regular point x̃ as reduction. We can
find a local parameter of x̃, i.e. a function f ∈ OX̃ (Ũ ) which has only one zero
on Ũ in x̃ and ordx̃ f˜ = 1. A lift f of f˜ can only have a single zero in X+ (x̃)
and has |f (y)| < 1 for all y ∈ X+ (x̃) according to Corollary 2.2.7. Therefore
f − f (x) ∈ OX (U ) solves the divisor [x] locally on U = π −1 (Ũ ).
Proposition 4.4.8. Let x1 , x2 ∈ Xi := π −1 (X̃i \ Sing X̃) be two points on
the same component of X. Then the line bundle L corresponding to DL =
×
[x1 − x2 ], is represented by an element fL = (fi,j ) ∈ H 1 (X, OX
) that is already
×
1
in H (X, O̊X ).
Proof. The set U1 = X \ π −1 ({x̃1 , x̃2 }) is formal open on X and since L is
trivial on U1 it can be represented by the constant function m1 = 1. Since L
has no support outside of Xi , we can choose a formal open covering U2 , . . . , Ur
of Xi and meromorphic functions mi on the Ui which represent DL by Lemma
4.4.7. Without restriction, we can set the absolute value of mi to 1, since the
supremum norm is multiplicative on Xi . Therefore the line bundle L will be
represented by (mi ) and the transformation functions fi,j = mj /mi on Ui ∩ Uj
×
have absolute value 1. Thus f ∈ H 1 (X, O̊X
).
×
Corollary 4.4.9. The analytic group J¯ is a subgroup of H 1 (X, O̊X
).
61
Proposition 4.4.10. The map
η : Jˆ → Jac X ; (g, j) 7→ g · j
is well-defined and a surjective morphism of analytic groups.
Proof. In order to show that the morphism η is well defined, we need to show
that every toric line bundle M has degree zero. But the connected variety Gtm,K
parameterizes the toric line bundles, so their degree cannot change and the
trivial bundle has degree zero. By Corollary 4.4.9 we know that J¯ is a subgroup
×
×
×
of H 1 (X, O̊X
) and the canonical map H 1 (X, O̊X
) → H 1 (X, OX
) coincides with
¯
the natural embedding J → Jac X for bundles of degree zero. The equivalence
relation ψ̄(g) = j −1 for g ∈ Ḡtm,K is corresponding to the fact that H 1 (X, R× )
×
is a subgroup of both H 1 (X, K × ) and H 1 (X, O̊X
).
By Lemma 4.4.6 we can write any line bundle L of degree zero as a product
of a toric bundle M and a formal bundle N . As we already noted, the toric
bundle M has degree zero which implies that N has degree zero as well and is
¯ Hence η is surjective.
in J.
Note. The morphism η is not injective as there are line bundles that are both
formal and toric and yet not induced by R× as we will see later in this chapter.
We can assume without restriction that the dual graph contains no loops, as
we have discussed in the introduction of this section. This leaves only the next
proposition as second case to evaluate.
Proposition 4.4.11. Let e ∈ E be an edge with corresponding double point
x̃, lying on components X̃i and X̃j , where X̃i corresponds to the source of e
and X̃j corresponds to the target of e. Let x1 ∈ π −1 (X̃i \ Sing X̃) and x2 ∈
π −1 (X̃j \ Sing X̃). We denote by q the height of the annulus of the formal fiber
of x̃.
Then the line bundle L with DL = [x1 − x2 ] can be factored into α · g with
α = α(e, q) ∈ H 1 (X, K × ), the weighted cocycle of q in e as in Definition 4.1.6
×
and g ∈ H 1 (X, O̊X
).
Proof. Let U1 be a formal neighborhood of π −1 (x̃). We have seen in Chapter 2
that the formal fiber of x̃ is an annulus with parameter ζ such that |ζ(x)| = 1
for x ∈ Xi and |ζ(x)| = q for x ∈ Xj . By proposition 4.4.8 we can adjust x1
and x2 on their respective component so that both x1 and x2 are in U1 . Then
L can be represented on the interior of U1 by the meromorphic function
m1 =
ζ − x1
,
ζ − x2
which expands uniquely to U1 . On U2 := X \π −1 ({x̃, x̃1 , x̃2 }) we choose m2 = 1.
Then U = {U1 , U2 } is a formal open covering of X and the mi represent DL .
m1 has absolute value 1 on Xi and absolute value q −1 on Xj . So m2 /m1 has
absolute value 1 on Xi and q on Xj . Refining U as in Lemma 4.4.6 therefore
gives us the toric line bundle α(e, q) in the factorization of f .
62
Recall that we have defined
ϕ : Jˆ → Rt
by
ϕ(a, g) = − log|a| .
To further describe η we need to look at its kernel. As it turns out this kernel
is a lattice in Jˆ i.e. a discrete subgroup. Such a group is mapped to a lattice
in Rt under ϕ, retaining most of the important information. The next theorem
uses the previous two propositions to give a basis for that lattice. In the proof
we will also construct the kernel of η without applying ϕ but this construction
is rather abstract, since the formal factor is left implicit.
Theorem 4.4.12. The group ϕ(ker(η)), the projection of first factor of the
kernel of the morphism η of Corollary 4.4.10 is a lattice of rank t in Rt . If the
simple cycles of Proposition 4.1.4 are denoted by γi , then this lattice is generated
by vi = (vij )tj=1 with
X
vij =
−d(e) · log|q(e)|
e∈γi ∩γj
where q(e) is the height of the formal fiber of the double point corresponding
to e and d(e) = 1 if e has the same orientation in γi as in γj and d(e) = −1
otherwise.
Proof. We factorize the trivial line bundle in a non trivial way for each cycle
γ = (v1 , . . . , vm ), vm+1 = v1 in G(X). Select a point xj on
N each component
∼
Xj := π −1 (X̃j \ Sing X̃) corresponding to vj . We know that m
k=1 [xk − xk+1 ] =
O. So by using Proposition 4.4.11 we get a toric factor M = (αij ). The
corresponding element a = (ai ) ∈ Gtm,K of α is easily computed with Lemma
4.1.7 to be
Y
ai =
q(e)d(e)
e∈γi ∩γ
where q(e) is the height of the formal fiber of the double point corresponding
to e and d(e) is the multiplicity of e in γ counted positive if the direction of e
in γ and γi coincide, negative if not. Note that the cycles γi are simple, so e is
only found once in γi .
Since every cycle can be written as a formal sum of the γi , each factorization
obtained by this procedure will have an image in the given lattice.
¯ = 0 and η −1 (J)
¯ ∩ J¯ = J,
¯ since ϕ was
Furthermore we obviously get ϕ(J)
¯
defined this way and η is an isomorphism when restricted to J. So ϕ(ker η) is
a lattice and we can concentrate on the first factor of every factorization of the
trivial line bundle.
Suppose there is another factorization O = M ⊗ N of the trivial line bundle
into a toric line bundle M and a formal line bundle N . We represent these
factors by the cocycles (αij ) and (gij ) on a formal open covering U as in 4.4.6,
i.e. U refines Z and Ui ∩ Uj ⊂ Xk := π −1 (X̃k \ Sing X̃). This means that for
×
each Ui ∈ U, we have an fi ∈ OX
(Ui ) with αij gij = fj /fi on Ui ∩ Uj .
63
We have two cases to consider. There are Ui for which Ui ⊂ Xk holds for
some k. In this case we can set |fi | = 1, changing α only by a coboundary.
The other case we have to consider is Ui ⊂ Ze with non empty intersection
with both components of Ze . For each e ∈ E there is only one such Ui since
we have chosen the refinement of U accordingly. Denote by X̃k the component
corresponding to the source of e and by X̃j the component corresponding to
the target of e. We can adjust fi such that |fi |Xk | = 1. Then f˜i , the reduction
of fi on X̃k , is well defined. Set ne := ordx̃ f˜i the order of f˜i in x̃ the double
point corresponding to e. We have |fi |Uj | = q ne .
Select an arbitrary point of non singular reduction xk on Xk and xj on Xj .
By Proposition 4.4.11 the line bundle Le corresponding to the divisor class
De = [−ne xk + ne xj ] will yield the same toric factor on Ui . This means that
O
L=
Le
e∈E
can be factored to L = M ⊗ N 0 .
0
¯
¯
PWe have O, N , N ∈ J by definition, hence M, L ∈ J. The divisor D =
e∈E De has support in the non-singular locus of the reduction. We can modify
D with the divisor of a rational function such that deg D|Xk = 0 holds for each
component X̃k . Modifying D by the divisor of a rational function does not
change the cocycles, since a rational function
Pcorresponds to the unit cocycle
in both factors. This means that the chain e∈E −ne e has zero boundary on
G(X) and is a cycle. Therefore ϕ(M) is already an element of the lattice defined
above.
Proposition 4.4.13. In the case where the sequence (4.4) splits, the lattice of
Theorem 4.4.12 lies completely in the Gtm,K factor of Jˆ i.e. we get
Jac X = Gtm,K /M × B
with B being the analytic group variety with good reduction of sequence (4.4)
Proof. We have the following diagram of exact sequences.
1
1
1
/ Ḡt
m,K
/M
1
! {
/ J¯
/ Jˆ
σ
η
/B
/1
! / Jac X
/1
Now the injective splitting morphism σ of (4.4) implies an arrow to Jac X which
is necessarily injective. Furthermore, in the splitting case we have J¯ = Ḡtm,K ×B
and therefore Jˆ = Gtm,K × B by its definition. Thus the kernel of η is restricted
to Gtm,K × {0}, which gives us a way to see M as a lattice purely in Gtm,K .
We want to calculate the abelian variety B in a specific example. For this we
need the following lemma.
64
Lemma 4.4.14. Let X and Y be two projective smooth curves and ρ : X → Y
be an algebraic surjective map of degree n coprime to char K. Then ρ∗ ◦ ρ∗ =
n · idJac Y , where ρ∗ is the induced map and ρ∗ := (ρ∗ )∗ its dual.
Proof. The situation is as follows
X
ρ
Y
JacO X
ρ∗
Jac Y
∼
/ (Jac X)0
ρ∗
/ (Jac Y )0
.
P
Let us look at a more explicit description of ρ∗ and ρ∗ . If D =
y∈Y ny y
P
∗
−1
is a divisor, then ρ D = y∈Y ny ρ (y). This definition coincides with the
one of Chapter 3 as one can
Peasily check by writing D as a Cartier divisors.
for all x1 , x2 ∈ X with
Similarly for Divisors D = x∈X nx x with nx1 = nx2 P
ρ(x1 ) = ρ(x2 ) the map ρ∗ (D) is given by ρ∗ (D) =
x∈X nx ρ(x). On can
check this once again by calculating (ρ∗ )∗ using cocylces and the fact that
according to Torelli’s theorem the map ρ∗ maps the theta structure of Jac Y
onto that of Jac X. Clearly ρ∗ ◦ ρ∗ = n · id. Another way to put this is
that ρ∗ ρ∗ L = N (ρ∗ (L)) = L⊗n where N is the norm induced by ρ on the
corresponding function fields.
Proposition 4.4.15. Let X be an analytic curve of genus g with semi-stable
reduction X̃ such that the dual graph of X̃ has cyclomatic number 1. Let furthermore ρ : X → Y be an algebraic surjective map of degree n such that Y
has a reduction whose dual graph is a tree and genus g − 1. Then the analytic
variety B of (4.4) is isogenous to Jac Y and J¯ is the total fiber space of an
n-torsion point of Pic0 (B).
Proof. The dual of the map ρ∗ is ρ∗ : Jac X → Jac Y and has a kernel of
dimension one. The reduced subscheme of this kernel is an elliptic curve E with
multiplicative reduction. We get Ē = Ḡm,K and a morphism ψ̄ : Ḡm,K → J¯
from the embedding. We know there is only one way up to automorphism to
embed Ḡm,K in J¯ so ψ̄ is the ψ̄ of Theorem 4.4.2.
Let ρ̄ : J¯ → B be the cokernel of ψ̄. Since ρ∗ ◦ ψ̄ = 0 we get a homomorphism
γ : B → Jac Y . Since Jac Y and B have the same dimension g − 1 and since
ρ∗ = γ ◦ ρ̄ we see that γ is an isogeny.
We construct σ : B → J¯ as σ = ρ∗ ◦ γ. With Lemma 4.4.14 we calculate
γ ◦ ρ̄ ◦ σ = γ ◦ ρ̄ ◦ ρ∗ ◦ γ
= ρ∗ ◦ ρ∗ ◦ γ
= n · idJac Y ◦γ .
so ρ̄ ◦ σ = n · idB . This means that for a rational section s : B 99K J¯ of ρ̄ we get
ns = σ up to coboundries. Therefore J¯ is the total fiber space of an n-torsion
point of Pic0 (B).
Note. It should be noted that even if B = Jac Y is the Jacobian of some curve,
the existence of the map ρ̄ : J¯ → B does not imply a map ρ : X → Y , since the
65
¯ In fact it is easy to construct an
curve X cannot be embedded in the variety J.
example where X has the same reduction properties as in Proposition 4.4.15
which does not come from an n-torsion point. Take any line bundle with good
reduction over any elliptic curve B. Dividing by a lattice gives a two dimensional
abelian variety which is the Jacobian of an genus two curve X by Torelli’s
Theorem. Since there are line bundles which are not n-torsion points there will
be no elliptic curve on which X can map onto.
4.5. Examples
Example 4.5.1. We will discuss Theorem 4.4.12 in the example of an elliptic
curve X. If X has good reduction the dual graph G(X) contains only one vertex
and has cyclomatic number t = 0. We therefore get J¯ = Jac X = X as usual.
In the case of bad reduction consider the model of Example 1.11.4. This means
√
that X̃ has two components of genus 0 meeting in two double points with | q|
as height of the formal fiber. We have exactly one simple cycle in G(X), which
√
√
means that our lattice is generated by the logarithm of |q| = | q 1 · q 1 |. Since
J¯ = Ḡm,K the sequence (4.4) splits and thus we know that Jˆ = Gm,K and that
there is a point q ∈ Gm,K such that all trivial toric bundles are represented by
q Z . And indeed Gm,K /q Z is equal to Jac X = X.
Example 4.5.2. Let us calculate the lattice for curves of genus 2. In case of
good reduction we have Jac X = J¯ as before. For bad reduction, we have three
cases to consider.
First we can have t = 0 as cyclomatic number. In this case the reduction of
X consists of two elliptic curves meeting at one ordinary double point. Since
t = 0 the toric part is trivial and Jac X = J¯ = B. Describing B and its relation
to the elliptic curves of the reduction in this case is rather difficult, some cases
are described in [FK91].
Second if the cyclomatic number t equals 1, then X̃ consists of an elliptic curve
intersecting a rational curve in two double points of height q1 and q2 respectively.
There is only one simple cycle and we see that the lattice is generated by
log|q1 q2 |. In the splitting case we furthermore get Jac X = Gm,K /(q1 q2 )Z × E
with an elliptic curve E with good reduction.
The lattice is the most interesting in the total degenerate case of t = 2 where
X̃ consists entirely of rational curves. Since B = 0 we know that the lattice lies
completely in G2m,K . There are two possibilities for G(X) save for blow-ups.
In the first case G(X) will look like
e1
•h
e2
(
e3
•h
(
• ,
e4
with corresponding heights q1 , q2 , q3 and q4 . We select γ1 = e1 + e2 and γ2 =
e3 +e4 . These cycles have no intersection, so the lattice is generated by (q1 q2 , 1)
and (1, q3 q4 ).
66
4.5. Examples
In the second case we get the graph G(X)
e1
• oe
e2
%
• ,
e3
with the height of the double points being q1 , q2 and q3 . We select the simple
cycles γ1 = e1 + e2 and γ2 = e1 + e3 . γ1 and γ2 intersect at e1 and e1 has
the same direction in both cycles. This means that the lattice is generated by
(q1 q2 , q1 ) and (q1 , q1 q3 ).
Example 4.5.3. We want to describe B in detail in the case of a special genus
two curve with cyclomatic number 1.
For this let λ, a, b ∈ K with |λ| < 1 and |a| = |b| = 1. We want to describe
the Jacobian of the genus two curve X given by
U1 := Sp Khξ, ηi/(η 2 − (ξ 2 − λ2 )(ξ 2 − a2 )(ξ 2 − b2 ))
U2 := Sp Khσ, τ i/(τ 2 − (1 − λ2 σ 2 )(1 − a2 σ 2 )(1 − b2 σ 2 ))
glued via σ = 1/ξ and τ = η/σ 3 .
The reduction induced by Ui is given by
Ũ1 := Spec k[X, Y ]/(Y 2 − X 2 (X 2 − ã2 )(X 2 − b̃2 ))
Ũ2 := Spec k[S, T ]/(T 2 − (1 − ã2 S 2 )(1 − b̃2 S 2 ))
with S = 1/X and T = Y /S 3 . This curve has a self intersection at X = 0, Y =
0. To get rid of this self intersection we blow U1 up by subdividing it in parts
|ξ| ≤ λ and |ξ| ≥ λ. We get
V1 := Sp Khχ, ϑi/(ϑ2 − (χ2 − 1)(λ2 χ2 − a2 )(λ2 χ2 − b2 ))
V2 := Sp Khξ, η, ζi/(ζξ − λ, η 2 − (1 − ζ 2 )(ξ 2 − a2 )(ξ 2 − b2 ))
V3 := Sp Khσ, τ i/(τ 2 − (1 − λ2 σ 2 )(1 − a2 σ 2 )(1 − b2 σ 2 ))
where χ = λ−1 ξ and η is modified accordingly.
This gives the reduction X̃
Ṽ1 := Spec k[V, W ]/(W 2 − (V 2 − 1))
Ṽ2 := Spec k[X, Y, Z]/(XZ, Y 2 − (1 − Z 2 )(X 2 − ã2 )(X 2 − b̃2 ))
Ṽ3 := Spec k[S, T ]/(T 2 − (1 − ã2 S 2 )(1 − b̃2 S 2 ))
which is the elliptic curve given by the ramification points ±ã and ±b̃ glued
with a rational curve intersecting at X = 0, Y = ±ãb̃. Any divisor with support
in the non singular locus of the reduction is a divisor with support in V1 ∪ V2 .
Since J¯ is part of the Jacobian which is birational to X (2) we can deduce that
ι : V2 (λξ −1 , λ−1 ξ) × V2 (ξ −1 ) −→ J¯
((ξp , ηp ), (ξq , ηq )) 7−→ [(ξp , ηp ) + (ξq , ηq ) − (λ, 0) − (a, 0)]
67
¯
is a formal birational map describing J.
There are two elliptic curves E1 and E2 gained by
U1 := Sp Khξ1 , η1 i/(η12 − ξ1 (ξ1 + λ2 )(ξ1 + a2 )(ξ1 + b2 ))
U2 := Sp Khσ1 , τ1 i/(τ12 − (1 + σ1 λ2 )(1 + σ1 a2 )(1 + σ1 b2 ))
and
V1 := Sp Khξ2 , η2 i/(η22 − (ξ2 − λ2 )(ξ2 − a2 )(ξ2 − b2 ))
V2 := Sp Khσ2 , τ2 i/(τ22 − σ2 (1 − σ2 λ2 )(1 − σ2 a2 )(1 − σ2 b2 ))
with σi = 1/ξi and τi = ηi /ξi2 as gluing relations. We also get maps of degree 2
ψ : X → E1
ρ : X → E2
ψ(ξ, η) = (−ξ 2 , η · ξ)
ρ(ξ, η) = (ξ 2 , η)
These maps induce pullbacks on Pic which give the maps
ψ ∗ : E1 → Jac X
[(−ξp2 , ηp ) − (−a2 , 0)] 7→
[(ξp , ηp /ξp ) + (−ξp , −ηp /ξp ) − (a, 0) − (−a, 0)]
∗
ρ : E2 → Jac X
[(ξp2 , ηp ) − (a2 , 0)] 7→
[(ξp , ηp ) + (−ξp , ηp ) − (a, 0) − (−a, 0)]
ψ∗ : Jac X → E1
[(ξp , ηp ) + (ξq , ηq ) − (a, 0) − (−a, 0)] 7→
[(−ξp2 , ξp ηp ) + (−ξq2 , ξq ηq ) − 2(−a2 , 0)]
ρ∗ : Jac X → E2
[(ξp , ηp ) + (ξq , ηq ) − (a, 0) − (−a, 0)] 7→
[(ξp2 , ηp ) + (ξq2 , ηq ) − 2(a2 , 0)] ,
where ψ∗ and ρ∗ are the dual maps of ψ ∗ and ρ∗ with {(ξp , ηp ), (ξq , ηq )} as the
preimage of ψ ∗ (x) or ρ∗ (x), extended to Jac X uniquely.
As noted in Lemma 4.4.14 we have
ψ∗ ψ ∗ ([(−ξp2 , ηp ) − (−a2 , 0)]) = ψ∗ ([(ξp , ηp /ξp ) + (−ξp , −ηp /ξp )
− (a, 0) − (−a, 0)])
=
[2(−ξp2 , ηp )
− 2(−a2 , 0)]
ρ∗ ρ∗ ([(ξp2 , ηp ) − (a2 , 0)]) = ρ∗ ([(ξp , ηp ) + (−ξp , ηp ) − (a, 0) − (−a, 0)])
= [2(ξp2 , ηp ) − 2(a2 , 0)] .
One checks that
ρ∗ ψ ∗ ([(−ξp2 , ηp ) − (−a2 , 0]) = [(ξp2 , ηp /ξp ) + (ξp2 , −ηp /ξp ) − 2(a2 , 0)] = 0 .
68
4.5. Examples
Furthermore ψ ∗ is injective and ρ∗ is surjective so we get the exact but not
strict exact sequence
/ E1
0
ψ∗
/ Jac X ρ∗
/0 .
/ E2
(4.7)
To see that (4.7) is not strict exact, look at the fibers of ρ∗ . The points [(ξp , ηp )+
(ξq , ηq ) − (a, 0) − (−a, 0)] that map to zero are given by ξp2 − ξq2 and ηp + ηq on
X 2 . This is a singular variety in Jac X so ρ∗ is not smooth according to [Har77,
III. Lemma 10.5]. The morphism ψ ∗ however is indeed a closed immersion.
The curve E1 has multiplicative reduction. Blowing up at |ξ| = |λ2 | again
resolves the self intersection and leaves us with Ē1 as the points of E1 with
a ξ coordinate with absolute value 1 by mapping (−ξp2 , ηp ) to the line bundle
[(−ξp2 , ηp ) − (−a2 , 0)]. The map ψ ∗ maps this line bundle to the line bundle
¯
[(ξp , ηp /ξp ) + (−ξp , −ηp /ξp ) − (a, 0) − (−a, 0)] which lies in J.
We know by [BGR84, 9.7.] that there is an element q ∈ Gm,K with |q| < 1
and an isomorphism ϕ : Gm,K /q Z → E1 which restricts to ϕ̄ : Ḡm,K → Ē1 . Then
ψ ∗ ◦ ϕ̄ : Ḡm,K → Jac X has its image in J¯ and is therefore a closed immersion
¯ Since there is up to automorphism of Ḡm,K only one injective
ψ̄ : Ḡm,K → J.
map from Ḡm,K to J¯ we can without restriction assume that ψ̄ is the ψ̄ of
Theorem 4.4.2.
Let ρ̄ : J¯ → B be the cokernel of this morphism. Since ρ∗ ◦ ψ ∗ = 0 and ψ̄ is a
restriction of ψ ∗ we get ρ∗ ◦ ψ̄ = 0 and therefore a morphism γ : B → E2 . Both
B and E2 have dimension 1 and since ρ∗ = γ ◦ ρ̄ we see that γ is surjective and
is therefore an isogeny.
We can directly calculate that B̃ is Spec k[X, Y ]/(Y 2 − (X 2 − ã2 )(X 2 − b̃2 ))
which is isogenous of order two to Ẽ2 .
Furthermore we know that ρ∗ ◦ ρ∗ = 2 · id and that we get the diagram
Ḡm,K
E2
ρ∗
ψ̄
/ J¯
" / Jac X
ρ̄
B
One calculates that ψ̄(Ḡm,K ) ∩ ρ∗ (E2 ) consists of 0 and the two torsion point
[(b, 0)+(−b, 0)−(a, 0)−(−a, 0)], So γ is an isogeny of order two and B is obtained
by completing Sp Khξ, ηi/(η 2 − (ξ 2 − (a2 − λ2 )(b2 − λ2 )) in the canonical way.
¯ Since
Let’s set σ = ρ∗ ◦ γ : B → J.
ψ̄
Ḡm,K
/ J¯
ρ̄
/B
ρ∗
γ
∗
/ E2 ρ / J¯
6
ρ̄
/B
γ
/ E2
6
ρ∗
we know that ρ̄ ◦ σ = ρ̄ ◦ ρ∗ ◦ γ : B → B is an isogeny. And since ρ∗ ◦ ρ∗ = 2 · id
69
we get
γ ◦ ρ̄ ◦ σ = γ ◦ ρ̄ ◦ ρ∗ ◦ γ
= ρ∗ ◦ ρ∗ ◦ γ
= 2 · idE2 ◦γ .
thus ρ̄ ◦ σ = 2 · idB .
Let s : B 99K J¯ be a rational section of ρ̄. Then 2s − σ : U → J¯ is in the image
of ψ̄ by the exactness of (4.4) and therefore 2s and σ only differ by a cocycle.
But σ is a group homomorphism and therefore defining a trivial bundle. This
means that J¯ is the total space of a two torsion point of Pic0 (B).
70
A. Bibliography
[BGR84] S. Bosch, U. Güntzer, and R. Remmert. Non-Archimedean analysis, volume 261 of Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences]. Springer-Verlag,
Berlin, 1984. A systematic approach to rigid analytic geometry.
[BL84]
Siegfried Bosch and Werner Lütkebohmert. Stable reduction and
uniformization of abelian varieties. II. Invent. Math., 78(2):257–297,
1984.
[BL85]
Siegfried Bosch and Werner Lütkebohmert. Stable reduction and
uniformization of abelian varieties. I. Math. Ann., 270(3):349–379,
1985.
[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron
models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. SpringerVerlag, Berlin, 1990.
[Bos05]
Siegfried Bosch. Lectures on formal and rigid geometry. In SFB 478 –
Geometrische Strukturen in der Mathematik, volume 378. University
of Münster, 2005.
[FK91]
Gerhard Frey and Ernst Kani. Curves of genus 2 covering elliptic
curves and an arithmetical application. In Arithmetic algebraic geometry (Texel, 1989), volume 89 of Progr. Math., pages 153–176.
Birkhäuser Boston, Boston, MA, 1991.
[Har77]
Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York,
1977. Graduate Texts in Mathematics, No. 52.
[LT58]
Serge Lang and John Tate. Principal homogeneous spaces over
abelian varieties. Amer. J. Math., 80:659–684, 1958.
[Mil86]
J. S. Milne. Jacobian varieties. In Arithmetic geometry (Storrs,
Conn., 1984), pages 167–212. Springer, New York, 1986.
[Mum08] David Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics. Published for the Tata
Institute of Fundamental Research, Bombay, 2008. With appendices
by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second
(1974) edition.
[Ray70]
Michel Raynaud. Anneaux locaux henséliens. Lecture Notes in Mathematics, Vol. 169. Springer-Verlag, Berlin, 1970.
71
A. Bibliography
[Ser88]
72
Jean-Pierre Serre. Algebraic groups and class fields, volume 117 of
Graduate Texts in Mathematics. Springer-Verlag, New York, 1988.
Translated from the French.
Ehrenwörtliche Erklärung
Hiermit erkläre ich, dass ich diese Arbeit selbstständig angefertigt habe und
keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie die
wörtlich oder inhaltlich übernommenen Stellen als solche kenntlich gemacht und
die Satzung der Universität Ulm zur Sicherung guter wissenschaftlicher Praxis
in der jeweils gültigen Fassung beachtet habe.
Ulm, der
. Juni 2013
Sophie Schmieg
73
Curriculum vitae
Persönliche Daten
Name:
Sophie Schmieg
Geburtsdatum:
4. September 1985
Geburtsort:
Ebersberg
Staatsangehörigkeit:
deutsch
Ausbildung
seit April 2009
Promotion in Mathematik an der Universität Ulm im
Institut für Reine Mathematik
März 2009
Diplom in Mathematik mit Nebenfach Informatik
(Thema: Zur Theorie der kompakten Riemannschen
Flächen und ihrer Jacobi-Varietäten)
Oktober 2006
Vordiplom in Mathematik mit Nebenfach Informatik
seit Oktober 2004
Studium der Mathematik mit Nebenfach Informatik
an der Universität Ulm
Juni 2004
Abitur am Gymnasium Grafing
Sophie Schmieg
75

Rigid analytic curves and their Jacobians - OPARU

Transcription

Similar documents

Education Positions and Research Experience - Analysis

Ausgabe Nr.33 / 2014