The Manin-Mumford Conjecture for number fields Danny

The Manin-Mumford Conjecture
for number fields
Danny Scarponi
Universite´ Toulouse III - Paul Sabatier
IMT - Equipe Picard
Advisor:
Prof.
Damian Rossler
The statement
4. If Y /R is a smooth projective curve of genus g ≥ 2, then the
torsor Y01 → Y0 is not trivial.
Manin-Mumford Conjecture for number fields is a deep and
important finiteness question (raised independently by Manin
and Mumford) regarding the intersection of a curve with the torsion
subgroup of its Jacobian:
¯ an algebraic closure
Theorem 1. Let K denote a number field, K
5. Since H 1(C0, FC∗0 Ω∨C0/k ) ' Ext1(FC∗0 ΩC0/k , OC0 ), the torsor C01 corresponds to an extension
of K and let C/K be a curve of genus g ≥ 2. Denote by J the
Jacobian of C and fix an embedding C ,→ J defined over K. Then
¯ ∩ J(K)
¯ tors is finite.
the set C(K)
The non-triviality of the torsor implies that this extension is nonsplit and one can deduce that the vector bundle E is ample.
T
HE
Theorem 1 was proved by Raynaud in 1983, see [Ray83a].
Some months later, Raynaud generalized his result obtaining
the following ([Ray83b]):
¯ be as above. Let A be an abelian variety
Theorem 2. Let K and K
and X an algebraic subvariety, both defined over K. If X does not
contain any translation of an abelian subvariety of A of dimension
¯ ∩ A(K)
¯ tors is finite.
at least one, then X(K)
Various other proofs (sometimes only for the case of curves)
later appeared, due to Serre ([Ser86]), Coleman ([C+87]),
Hindry ([Hin88]), Buium ([Bui96]) , Hrushovski ([Pil97]), PinkRossler ([PR02]) and Baker-Ribet ([BR02]).
N
6. C01 identifies with P(E)\P(FC∗0 ΩC0/k ) which is affine thanks to the
ampleness of E.
7. If p > 2, the map ∇10 : J(R) → J01(k) is injective if restricted to
J(R)tors, so
](C(R) ∩ J(R)tors) = ](∇10(C(R) ∩ J(R)tors)).
8. Let B := pJ01 be the maximal abelian subvariety of J01. Then the
image of ∇10(J(R)tors) under the homomorphism
J01(k) → J01(k)/B(k)
is a finite set.
9. ∇10(C(R) ∩ J(R)tors) is a finite union of sets of the type
Buium’s proof
OW
0 → OC0 → E → FC∗0 ΩC0/k → 0.
(B(k) + b) ∩ C01(k),
we sketch Buium’s proof of Theorem 1:
1. Thanks to a result due to Coleman (see [C+87]), Theorem 1 is
an easy consequence of its “non-ramified version”:
Theorem 3. Let k be an algebraically closed field of characteristic p > 0 and let R be the ring of Witt vectors with coordinates
in k. Let C/R be a smooth projective curve of genus g ≥ 2 possessing an R-point and embedded via this point into its Jabobian J/R. Then C(R) ∩ J(R)tors is finite.
2. For any variety Y over R and for any n ∈ N, Buium defines the
p-jet space of Y1 := Y ⊗R R/p2 of order n.
3. The first order p-jet space of Y1 is denoted Y01 and is provided with a map ∇10 : Y (R) → Y01(k). If Y is smooth along
Y0 := Y ⊗R k, then Y01 is a torsor on Y0 under the Frobenius
tangent bundle F T (Y0/k) := Spec(SymFY∗0 ΩY0/k ).
where b ∈ J01(k). Each of these is finite, since B is proper and
C01 is affine.
My research
aim is to generalize Buium’s proof to any dimension in order
to give a new proof of Theorem 2. As in the dimension one
case, it is still true that the main point is proving the non-ramified
version. So let R be as before. Let A be an abelian variety and X
an algebraic subvariety both defined over R. The main difficulty in
generalizing Buium’s work are parts (5)-(6): if X is a subvariety of
dimension greater than one, then X01 needs not to be affine. The
idea to overcome this problem consists in regarding not only X01,
but also the higher p-jet spaces and in using some tools coming
from the theory of strongly semistable sheaves.
M
Y
References
[BR02]
Matthew Baker and Kenneth A. Ribet. Galois theory and torsion points on curves. arXiv preprint math/0212133, 2002.
[Bui96]
Alexandru Buium. Geometry of p -jets. Duke Math. J., 82(2):349–367, 02 1996.
[C+87]
R. Coleman et al. Ramified torsion points on curves. Duke Math. J, 54(2):615–640, 1987.
[Hin88]
Marc Hindry. Autour d’une conjecture de Serge Lang. Inventiones mathematicae, 94(3):575–603, 1988.
[Pil97]
Anand Pillay. Model theory and diophantine geometry. Bulletin of The American Mathematical Society, 34(4):405–422, 1997.
[PR02]
Richard Pink and Damian Roessler. On Hrushovski’s proof of the Manin-Mumford conjecture. arXiv preprint math/0212408, 2002.
´ e´ abelienne
´
[Ray83a] Michel Raynaud. Courbes sur une variet
et points de torsion. Inventiones mathematicae, 71(1):207–233, 1983.
´ es
´ d’une variet
´ e´ abelienne
´
[Ray83b] Michel Raynaud. Sous-variet
et points de torsion. In Arithmetic and geometry, pages 327–352.
Springer, 1983.
[Ser86]
`
J.-P. Serre. Course at the college
de France. 1985-1986.
GAeL XXIII, Leuven, 2015