ROOT NUMBERS OF HYPERELLIPTIC CURVES 1. Introduction

Transcription

ROOT NUMBERS OF HYPERELLIPTIC CURVES
ARMAND BRUMER, KENNETH KRAMER, AND MARIA SABITOVA
1. Introduction
The root number W (A) associated to an abelian variety A over a number field L is the
sign in the (conjectural) functional equation of its L-function, and so the conjectures of
Birch and Swinnerton-Dyer imply that the rank of the Mordell–Weil group A(L) satisfies
W (A) = (−1)rank A(L) .
For each place v of L let Lv denote the completion of L with respect to v. Let W (Av ) be
the local root number associated to Av = A ×L Lv . By definition,
Y
W (A) =
W (Av ),
v
where v runs through all the places of L. If v|∞, then
W (Av ) = (−1)dim A
(see Proposition 1 in [Ro93] or by Lemma 2.1 on p. 4272 in [S07]). If v is finite, then
W (Av ) is defined as follows. Let Lv be a fixed algebraic closure of Lv . For a rational prime
l different from the residual characteristic of Lv let Tl (Av ) be the l-adic Tate module of
0
denote a (complex) representation of the
Av and let Vl (Av ) = Tl (Av ) ⊗Zl Ql . Let σv0 = σv,A
0
Weil–Deligne group W (Lv /Lv ) of Lv associated to Vl (Av ) via the Deligne–Grothendieck
construction (see e.g., [Ro94]). Then
W (Av ) = W (σv0 ).
We assume from now on a non-archimedean local field of definition K of A. Let AOK
and AOF be the Néron models over the rings of integers of K and a (possibly trivial)
Galois extension F/K such that A has semistable reduction over F . We analyze W (A)
in terms of contributions from the toroidal and abelian variety components of the closed
fibers of these Néron models. In particular, our results can be used to calculate local
root numbers of abelian varieties in two main instances: first, for abelian varieties with
additive reduction over K and totally toroidal reduction over F , provided that the residual
characteristic of K is odd and the wild part of the inertia group I(F/K) of F/K is abelian,
and second, for the Jacobian J(X) of a stable curve X over K. More precisely, in the
first case
W (A) = χ(−1)n ,
Date: March 19, 2015.
1
2
A. BRUMER, K. KRAMER, AND M. SABITOVA
where χ is a ramified quadratic character of K × and n is the number of irreducible
components of the complex representation of I(F/K) associated to (the toric part of) the
closed fiber of AOF (see (2.7), Proposition 2.2, and Proposition 3.2 below). In the second
case,
Y
τ[z] ,
W (J(X)) = (−1)nk +sk +1
[z]∈Σk
where k is the residue field of K, Σk is the set of the singular k-points of the reduction X
of X over k, nk = |Σk |, sk is the number of irreducible k-components of X and τ[z] = ±1
is defined in Proposition 4.2 below. We also give examples of hyperelliptic curves of genus
2 over Q, for which the obtained results are enough to calculate the global root numbers
of their Jacobians.
For a somewhat different viewpoint on epsilon factors associated to abelian varieties
which acquire semistable reduction over a tamely ramified extension of a local field, see
Lemma 1 in [Sa93].
The paper is organized as follows. Section 2 contains general facts and notation concerning root numbers and abelian varieties that are used in the paper. In Section 3 we
calculate the contribution to the local root number W (A) from the toric parts of the
closed fibers of AOK and AOF in the case when the residual characteristic of K is odd
and the wild part of the inertia group of F/K is abelian. In Section 4 we calculate the
root number of the Jacobian of a stable curve X over K in terms of the reduction of X
over K. Finally, Section 5 provides several examples of hyperelliptic curves of genus 2
over Q, for which the obtained results allow one to calculate global root numbers of their
Jacobians.
2. General facts and notation
2.1. Base field and representations of the Weil–Deligne group. Let K be a local
non-archimedean field with ring of integers OK and residue field k of characteristic p. Let
K and k be fixed algebraic closures of K and k, respectively. Let ϕ be the inverse of the
Frobenius automorphism of the absolute Galois group of k and let Φ ∈ Gal(K/K) denote
a preimage of ϕ under the decomposition map. We call a continuous homomorphism
µ : K × −→ C× a (multiplicative) character of K × . By definition, the Weil group W(K/K)
(also denoted by WK ) of K is a subgroup of Gal(K/K) equal to Gal(K/K unr )ohΦi, where
K unr ⊂ K denotes the maximal unramified extension of K contained in K, hΦi denotes the
infinite cyclic group generated by Φ, and IK = Gal(K/K unr ) is the inertia group of K. The
group W(K/K) is a topological group (see e.g., [Ro94]). Throughout the paper we will
identify one-dimensional complex continuous representations of W(K/K) with characters
of K × via local class field theory assuming that a uniformizer of K corresponds to a
geometric Frobenius Φ ∈ Gal(K/K). Let ω : W(K/K) −→ C× be the one-dimensional
representation of W(K/K) given by
ω|IK = 1,
ω(Φ) = q −1 ,
3
where q = card(k).
Let W 0 (K/K) denote the Weil–Deligne group of K. By a representation σ 0 of W 0 (K/K)
we mean a continuous homomorphism
σ 0 : W 0 (K/K) −→ GL(U ),
where U is a finite-dimensional complex vector space and the restriction of σ 0 to the
subgroup C of W 0 (K/K) is complex analytic. It is known that there is a bijection between
representations of W 0 (K/K) and pairs (σ, N ), where σ : W(K/K) −→ GL(U ) is a
continuous complex representation of W(K/K) and N is a nilpotent endomorphism on
U such that
σ(g)N σ(g)−1 = ω(g)N, g ∈ W(K/K).
In what follows we identify σ 0 with the corresponding pair (σ, N ) and write σ 0 = (σ, N ).
Also, a representation σ of W(K/K) is identified with the representation (σ, 0) of W 0 (K/K)
(see e.g., §§1–3 in [Ro94]).
2.2. Abelian varieties. Let A be an abelian variety over K. For a rational prime l
different from p = char(k) let Tl (A) be the l-adic Tate module of A. It is a free Zl -module
of rank 2g, where g = dim A. Put Vl (A) = Tl (A) ⊗Zl Ql and let
ρl : Gal(K/K) −→ GL(Vl (A))
denote the natural l-adic representation of Gal(K/K) on Vl (A). The Weil pairing together
with an isogeny from A to the dual abelian variety of A induces a nondegenerate, skewsymmetric, Gal(K/K)-equivariant pairing
(2.1)
h−, −i : Vl (A) × Vl (A) −→ Ql ⊗ ωl ,
where ωl is the l-adic cyclotomic character of Gal(K/K) (see e.g., [S07] for more detail).
Let F/K ⊂ K/K be a finite Galois extension over which A acquires semistable reduction ([G67], p. 21, Theorem 6.1). Denote by kF the residue field of F . Let A denote the
Néron model of A. Let AvK = A ×OK k and AvF = A ×OK kF be the closed fibers of
A over k and kF , respectively, with corresponding connected components A0vK and A0vF .
Let TvK and TvF be the maximal subtori of A0vK and A0vF , respectively. As an algebraic
group, A0vK has a natural decomposition
(2.2)
0 → U → A0vK /TvK → BvK → 0,
in which U is a unipotent algebraic group over k and BvK is an abelian variety over k.
Since A acquires semistable reduction over F ,
(2.3)
A0vF /TvF ∼
= BvF ,
where BvF is an abelian variety over kF .
4
We will use the following notation:
V
= Vl (A) = Tl (A) ⊗Zl Ql ,
V1 = V IK ,
V2 = V IF ,
so that V1 ⊆ V2 ⊆ V as Gal(K/K)-modules over Ql . The reduction map π : OK k
induces the following isomorphisms:
V1 ∼
= Tl (A0vK ) ⊗Zl Q,
(2.4)
V2 ∼
= Tl (A0vF ) ⊗Zl Ql
(as follows easily from [ST68], p. 495, Lem. 2). We endow Tl (A0vK ) and Tl (A0vF ) with the
action of Gal(K/K) via isomorphisms (2.4). It follows from the orthogonality theorem
([G67], p. 338, Theorem 2.4) that the reduction map π induces the following isomorphism
of Gal(K/K)-modules:
V1 ∩ V ⊥ ∼
(2.5)
= Tl (Tv ) ⊗Z Ql ,
1
where
V1⊥
K
l
is an orthogonal complement to V1 in V with respect to the pairing (2.1).
Proposition 2.1. We have
• V2⊥ ⊆ V2 ,
• V1 ∩ V1⊥ ⊆ V2⊥ .
Proof. This follows from Proposition 3.5 on p. 350 in [G67] and Equation (2.7) on p. 32
in [E95].
Note that (2.2) and (2.3) induce exact sequences
0 −→ Tl (TvK ) −→ Tl (A0vK ) −→ Tl (BvK ) −→ 0,
0 −→ Tl (TvF ) −→ Tl (A0vF ) −→ Tl (BvF ) −→ 0,
since T (k) is a divisible group for a torus T over k, so that lim is an exact functor, and
←−
Tl (U) is trivial. Thus from (2.5) and Proposition 2.1 we have the following diagrams
corresponding to each other via the reduction map
VO 1 ?

V1 ∩ V1⊥
/
/

Tl (A0vK ) VO 2
/
O
?
?
Tl (TvK ) V2⊥

Tl (A0vF )
/
O
?
Tl (TvF )
Since V1 ∩V2⊥ = V1 ∩V1⊥ by Proposition 2.1, this gives identification of Gal(K/K)-modules
V1 /(V1 ∩ V ⊥ ) ∼
= Tl (Bv ) ⊗Z Ql , V2 /V ⊥ ∼
= Tl (Bv ) ⊗Z Ql ,
1
K
l
2
F
l
as well as an injection of Gal(K/K)-modules
Tl (BvK ) ⊗Zl Ql ∼
= (Tl (A0vK )/Tl (TvK )) ⊗Zl Ql ,→ Tl (BvF ) ⊗Zl Ql ∼
= (Tl (A0vF )/Tl (TvF )) ⊗Zl Ql .
5
Let ρ¯tor denote the representation of WK on the quotient U = (Tl (TvF )/Tl (TvK )) ⊗Zl Ql
considered as a representation over C via a fixed embedding ı : Ql ,→ C:
ρ¯tor : WK −→ GL((Tl (TvF )/Tl (TvK ) ⊗Zl Ql ) ⊗ı C).
Let
H = IK /IF = Gal(F unr /K unr ).
Clearly, by (2.5) the action of IK on U factors through H and U H = {0}. Let ρ¯av denote
the representation of W(K/K) on the quotient of Tl (BvF ) by Tl (BvK ) considered as a
representation over C via ı:
ρ¯av : WK −→ GL(((Tl (BvF )/Tl (BvK )) ⊗Zl Ql ) ⊗ı C).
We are interested in the representation ρ0 = (ρ, N ) of W 0 (K/K) associated to ρl by
the standard procedure (see e.g., [Ro94], §4). More precisely, we fix a non-trivial homomorphism tl : IK −→ Zl , e.g., the one that factors through the natural projection
Gal(K l /K unr ) −→ Zl corresponding to the maximal pro-l extension K l ⊂ K of K unr .
Then there exists a unique nilpotent endomorphism Nl of V such that
ρl (h) = exp(tl (h)Nl ),
h ∈ IF .
We have ker Nl = V2 and the image of Nl is contained in V2⊥ , thanks to the perfect pairing
(2.1). Indeed,
h(ρl (h) − id)w, vi = 0, h ∈ IF , v ∈ V2 , w ∈ V.
Let ı : Ql ,→ C be a field embedding and let N = Nl ⊗ı C. Define a representation ρ of
W(K/K) on V ⊗ı C as
(2.6)
ρ(g) = ρl (g) exp(−tl (h)Nl ),
g = Φm h ∈ W(K/K), m ∈ Z, h ∈ IK .
Then ρ0 = (ρ, N ) is a representation of W 0 (K/K) on V ⊗ı C. It is known that in our
context ρ0 does not depend on the choice of l and ı (see e.g., [S07]). By definition,
(2.7)
W (A) = W (ρ0 ) = W (ρ) · δ(ρ0 ),
where
δ(ρ0 ) = det −Φ|(V ⊗ı C)IK /(ker N )IK
(see e.g., [Ro94], §§11, 12, for the definition and properties of W (ρ)).
Proposition 2.2. We have
W (ρ) = W (ρav ) · (det ρtor )(−1).
In particular, if A already is semistable over K, then W (ρ) = 1.
Proof. By additivity, W (ρ) is the product of the root numbers of the representations of
W(K/K) (over C via ı) afforded by the successive quotients in
0 ⊆ V2⊥ ⊆ V2 ⊆ V.
Since Nl ⊗Zl Ql acts as zero on each of these quotients, ρ(WK ) and ρl (WK ) agree on each
quotient.
6
By Proposition 2.1, the first constituent V2⊥ affords the representation
ρtor : W(K/K) −→ GL((Tl (TvF ) ⊗Zl Ql ) ⊗ı C)
and the second constituent V2 /V2⊥ affords the representation
ρav : W(K/K) −→ GL((Tl (BvF ) ⊗Zl Ql ) ⊗ı C).
The perfect pairing
V2⊥ × (V /V2 ) → Ql ⊗ ωl
shows that the representation ν of W(K/K) afforded by the third constituent V /V2 satisfies ν ∼
= (ρtor )∗ ⊗ ω. Hence, W (ρtor ⊕ ν) = (det ρtor )(−1) (see e.g., [Ro94], §§11, 12).
×
Any lift τ ∈ W(K/K) of −1 ∈ K ∼
= W(K/K)ab is in IK and therefore acts trivially on
the submodule V1 of V2 . Therefore, (det ρtor )(−1) = (det ρ¯tor )(−1).
Finally, the subrepresentation ρ0 : W(K/K) −→ GL((Tl (BvK ) ⊗Zl Ql ) ⊗ı C) of ρav is
unramified, so W (ρ0 ) = 1 by the standard properties of root numbers (see e.g., [Ro94],
§§11, 12). Hence, W (ρav ) = W (¯
ρav ).
3. Calculation of det ρ¯tor (−1)
Lemma 3.1. Let B be a finite abelian group of odd order, let C = hci be a finite cyclic
group of order h, and let G = B oC be a semi-direct product with C acting on B. Assume
that there exists a non-trivial orthogonal irreducible representation σ of G over C. Then
h is even. Moreover, if σ is not trivial on B, then there exists a subgroup Γ = B o hcx i
of G and a one-dimensional representation ψ of Γ such that
σ∼
= IndG
Γ ψ,
where x is even and ψ(cx ) = 1.
Proof. Clearly, if σ is a non-trivial orthogonal irreducible representation and σ is trivial
on B, then σ has order 2 and h is even. Thus, for the rest of the proof we assume that σ
is not trivial on B.
Using Mackey method of small subgroups, an irreducible representation σ of G can be
constructed from an irreducible (one-dimensional) representation ψ of B in the following
way. Let Γ denote the stabilizer of ψ in G, i.e.,
Γ = {g ∈ G | ψ(g −1 bg) = ψ(b), ∀b ∈ B}.
Then Γ = B o hcx i for some positive integer 0 < x ≤ h. We fix a choice of x such that
x divides h. It turns out that ψ can be extended to a representation of Γ. Namely, since
cx belongs to the stabilizer of ψ, we have ψ(c−x bcx ) = ψ(b) for any b ∈ B. For a (h/x)-th
root of unity ξ we can extend ψ to a representation of Γ via
ψ(cx ) = ξ.
7
By abuse of notation we will denote the extension of ψ to Γ also by ψ, so that
ψ(bcxv ) = ψ(b) · ξ v ,
b ∈ B, v ∈ {0, 1, . . . ,
h
− 1}.
x
Furthermore, σ = IndG
Γ ψ ([S77], p. 62, Prop. 25). Since by assumption, σ is not trivial
on B, ψ is not trivial.
We are interested in the case when σ is orthogonal. Then, in particular, ResG
B σ is
orthogonal and hence
ResG σ ∼
= 1n ⊕ µ ⊕ µ,
B
where µ is a representation of B and µ denotes its contragredient (recall that by assumption B is abelian of odd order). On the other hand,
∼
ResG
B σ = ψ ⊕ ψc ⊕ · · · ⊕ ψcx−1 ,
where ψci denotes the representation of B given by ψci (b) = ψ(c−i bci ), b ∈ B. Since
ψ is not trivial on B, we have n = 0 and ψ = ψcj as representations of B for some j.
Furthermore, one can easily check that
• x is even,
• ψ = ψcx/2 ,
• the character of ψ is not real-valued, i.e., ψ is neither trivial nor quadratic.
Recall that for an irreducible representation σ of a finite group G we have


if and only if σ is orthogonal,
1,
1 X
2
·
tr σ(y ) = −1, if and only if σ is symplectic,
(3.1)

|G| y∈G
0,
if and only if the character of σ is not real-valued
([S77], p. 109, Prop. 39). We now apply (3.1) to G = B o hci and σ = IndG
Γ ψ. We first
2
2
v
note that tr σ(y ) = 0 unless y ∈ Γ. Let y = bc for b ∈ B, v ∈ {0, 1, . . . , h − 1}, so that
y 2 ∈ Γ if and only if x divides 2v and, since x is even, if and only if x2 divides v. Let
v = x2 n, then we have
X
X
2
S
=
tr
σ(y
)
=
tr σ(bcv bcv ) = S1 + S2 ,
(3.2)
y∈G
b, n
where
S1 =
X
b∈B
n=even
tr σ(bcv bcv ),
S2 =
X
tr σ(bcv bcv ).
b∈B
n=odd
We first show that S1 = 0. Let W be a representation space of G corresponding to σ
and let V ⊆ W be a one-dimensional subrepresentation of ResG
Γ σ isomorphic to ψ. Then
8
W = V ⊕ cV ⊕ c2 V ⊕ · · · ⊕ cx−1 V and in a suitable basis σ has the following



ψ(b)
0
...
0
0 0 . . . ψ(cx )
 0

 1 0 ...
ψc (b) . . .
0
0
 , σ(c) =  . . .
σ(b) = 
.
.
.
..
.
 ..

 .. .. . .
..
..
..
.
0
0
. . . ψcx−1 (b)
0 0 ...
0
form


,

where ψcx/2 = ψ, ψcx/2+1 = ψ c , . . ., ψcx−1 = ψ cx/2−1 . Let v = x2 n, where n = 2l is even, so
that v = xl, l ∈ {1, 2, . . . , h/x}, and σ(cx ) = ψ(cx )I, where I is the x × x-identity matrix.
Hence,
X
X
X
S1 =
tr σ((bcxl )2 ) =
tr ψ(γ 2 ) +
tr ψc (γ 2 ) + · · · +
(3.3)
γ∈Γ
b∈B, l
+
X
γ∈Γ
tr ψcx−1 (γ 2 ) = x
γ∈Γ
X
tr ψ(γ 2 ).
γ∈Γ
Since by assumption the character of ψ considered as a representation of Γ is not realvalued, it follows from (3.1) that S1 = 0.
We now calculate S2 . Let n = 2l + 1, v = x2 n. Using the matrix representations of
σ(b) and σ(c) above and taking into account that ψcx/2 (b)ψ(b) = 1 for any b ∈ B, one can
easily check that σ((bcv )2 ) = ξ 2l+1 I. Finally,
(
h
−1
x
X
0, ξ 2 6= 1,
S2 = x · |B| ·
ξ 2l+1 = x · |B| · ξ · h
, ξ 2 = 1,
x
l=0
since ξ h/x = 1. Recall that S1 + S2 = S2 = |G| = h · |B|, which implies ξ = 1.
Analogously, σ is symplectic if and only if x is even and ξ = −1. These generalize the
results of Proposition 1.5 in [S07].
For the rest of the section we will use the notation of Section 2.2.
Proposition 3.2. If p 6= 2 and the p-Sylow subgroup of H = Gal(F unr /K unr ) is abelian,
then
(3.4)
(det ρ¯tor )(−1) = χ(−1)n ,
where χ is a ramified quadratic character of K × and n is the number of irreducible components of ρ¯tor considered as a representation of H (in what follows denoted by ρ¯tor |H ).
Proof. Let H = B o C, where C = hci is a finite cyclic group of order t not divisible by
p and B is a p-group, i.e., B is the wild subgroup of H and C is the tame quotient. Let
α ∈ IK correspond to −1 under the local class field theory homomorphism WK −→ K ×
and let α
¯ = βγ ∈ H, β ∈ B, γ ∈ C, be the image of α under the quotient map IK −→ H,
so that
(det ρ¯tor )(−1) = (det ρ¯tor )(α) = (det ρ¯tor )(¯
α).
9
If the order of H is odd (equivalently, the order of B is odd), then clearly α
¯ = 1 and
the left side of (3.4) is 1. On the other hand, using (¯
ρtor )H = {0} and Lemma 3.1, as an
orthogonal representation of H, ρ¯tor |H ∼
= µ ⊕ µ∗ , where µ is a representation of H and µ∗
denotes its contragredient. Thus, n is even and the right side of (3.4) is also 1.
Assume for the rest of the proof that the order of H is even. Since p 6= 2, this means
the order of B is even. Since W H = {0}, as an orthogonal representation of H,
ρ¯tor |H ∼
= µ ⊕ µ∗ ⊕ λ1 ⊕ · · · ⊕ λm ⊕ ν q ,
where µ is a representation of H, µ∗ denotes its contragredient, each λi is an irreducible
orthogonal representation of H with dim λi ≥ 2, and ν is the one-dimensional representation of H of order 2. Using Lemma 3.1 it is easy to see that det λi (c) = −1.
Since ρ¯tor is orthogonal and by assumption the order of B is odd, (det ρ¯tor )(β) = 1 for
any β ∈ B and hence
(
1,
γ ∈ C 2,
(3.5)
(det ρ¯tor )(−1) = (det ρ¯tor )(α) = (det ρ¯tor )(γ) =
(−1)m+q , otherwise.
Note that every two ramified quadratic characters of K × differ by multiplication by
the unramified quadratic character of K × , which makes χ(−1) well-defined. Let $K be
√
√
a uniformizer of K and let t $K ∈ K be a t-th root of $K . Then K1 = K( t $K ) is a
totally ramified tame extension of K. We can take a quadratic character χ of K × given
by
φ
χ : WK ,→ Gal(K/K) −→ Gal(K1 /K) ∼
= Gal(K1unr /K unr ) = C −→ Z/2Z,
where φ(c) = −1 (recall that the order of C is even) and the rest are the natural maps.
Then χ(−1) = χ(α) = φ(γ) and the proposition follows from (3.5).
4. Calculation of δ(ρ0 )
In this section we keep the notation of Section 2.2.
Proposition 4.1. Denote
VNIK = (ker(Nl ⊗Zl Ql ))IK ,
Vl (TvK ) = Tl (TvK ) ⊗Zl Ql .
There is a perfect pairing
V IK /VNIK × Vl (TvK ) → Ql ⊗ ωl .
Proof. Recall the notation V1 = V IK , V2 = V IF . To determine V IK , consider the filtration
V ⊇ V1⊥ + V2 ⊇ V2 ⊇ 0. Since ρ(IF ) = 1 (by the definition of ρ), the action of ρ(IK ) on
V factors through the finite group IK /IF ∼
= Gal(F nr /K nr ) and therefore is semisimple.
Hence, we find the ρ(IK )-invariants on the successive quotients, i.e., on
(4.1)
V /(V1⊥ + V2 ) ⊕ (V1⊥ + V2 )/V2 ⊕ V2 .
On each piece Nl ⊗Zl Ql acts trivially, so ρ(IK ) and ρl (IK ) give the same action.
10
The action of ρl (IK ) is trivial on the first quotient V /(V1⊥ + V2 ), because
(ρl (i) − id)(V ) ⊆ V1⊥ ,
i ∈ IK .
By abuse of language we will say that the action of a group on a vector space is fixed
point free if the only fixed point is 0. We claim that the action of ρl (IK ) on (V1⊥ + V2 )/V2
is fixed point free. It suffices to show that the action of ρl (IK ) on the dual space is fixed
point free. Note that via the pairing (2.1), we have
∗
(V1⊥ + V2 )/V2 ∼
= V2⊥ /(V2⊥ ∩ V1 ).
But the latter injects into V2 /V1 , where 0 clearly is the only invariant point.
As for the third term in (4.1), we have V2IK = V1 and hence
V IK ∼
= V /(V ⊥ + V2 ) ⊕ V1 .
1
IK
Finally, ker(Nl ⊗Zl Ql ) = V2 and so VN = V1 . It follows that
V IK /VNIK ∼
= V /(V1⊥ + V2 ).
The latter pairs perfectly with V2⊥ ∩V1 = V1⊥ ∩V1 , which is isomorphic to Vl (TvK ) by (2.5).
Moreover, the action of Φ ∈ Gal(K/K) on V IK /VNIK corresponds to that of ϕ ∈ Gal(k/k)
on Vl (TvK ).
Let
×
H(TvK ) = HomZ TvK (k , k )
be the character group of the torus TvK and recall that ϕ ∈ Gal(k/k) denotes the inverse
of the Frobenius automorphism of k. The natural Gal(k/k)-equivariant pairing
×
TvK k × H(TvK ) → k
induces a perfect pairing
Vl (TvK ) × HQl (TvK ) → Ql ⊗ ωl ,
where HQl (TvK ) = H(TvK ) ⊗Zl Ql . Proposition 4.1 therefore gives
δ(ρ0 ) = det −Φ|(V ⊗ı C)IK /(ker N )IK = det −ϕ|HQl (TvK )⊗ı C .
4.1. Root numbers of Jacobians of curves. In this subsection we assume that A is
the Jacobian J(X) of a smooth geometrically irreducible proper curve X over K such
that the reduction XvK of X is a stable curve over k in the sense of Definition 1.1 on p.
76 in [DM69]. Thus A = J(X) has semistable reduction over K (by Theorem 2.4 on p.
89 in [DM69]), i.e., in our notation above F = K. Under these conditions W (ρ) = 1 by
Proposition 2.2 and hence
W (J(X)) = δ(ρ0 ).
It is known that the connected component of the reduction of J(X) over k is the
Jacobian of the reduction of X, i.e., A0vK = J(XvK ) = Pic0 (XvK ) (by the results in [R70]).
To simplify the notation throughout this subsection we will write X and T instead of
11
XvK and TvK , respectively. Let η : X˜ → X be the normalization of X , let X˜1 , . . . , X˜s be
irreducible components of X˜ , and let Σ be the set of singular points of X over k with
n = |Σ|. There is the following long exact sequence
(4.2)
×
×
×
1 → k −→ ⊕si=1 k −→ ⊕ni=1 k → T (k) → 1
([BLR90], p. 247).
×
To describe the character group H(T ) = HomZ (T (k), k ), we construct a directed
graph Γ, as follows. The set of vertices V is the set of irreducible components of X˜ . For
each node of X , say z ∈ Σ, fix an ordering, say (x, y) for the pair of points over z on
X˜ . Suppose that x and y lie on the components X˜i and X˜j , respectively. Let there be
a directed edge ez of Γ from the vertex X˜i to the vertex X˜j . We allow the possibility
of a loop if i = j. Also, we define the boundary map γ(ez ) = X˜j − X˜i . Denote the set
of edges of Γ by E. Let C0 (Γ, Z) and C1 (Γ, Z) be the free abelian groups on V and E,
respectively. It follows from [BLR90], p. 247, that dualizing (4.2), we obtain the following
exact sequence of Gal(k/k)-modules:
(4.3)
γ
a
0 → H(T ) → C1 (Γ, Z) → C0 (Γ, Z) → Z → 0,
in which a is the augmentation map induced by a(X˜i ) = 1 for all i. Here Gal(k/k) acts
trivially on Z, the Galois action on C0 (Γ, Z) is induced from the Galois action on the
set of irreducible components C = {X˜1 , . . . , X˜s }, the Galois action on C1 (Γ, Z) is induced
from the action on ordered pairs (x, y) 7→ (g(x), g(y)) for g ∈ Gal(k/k), and Gal(k/k)
acts canonically on H(T ).
It is convenient to consider the set of singular k-points Σk of X . Thus each element
of Σk is an orbit, denoted [z], of a node z ∈ Σ under the action of Gal(k/k). Similarly,
define Ck , the set of irreducible k-components of X˜ . Let z1 and z2 be the points over z in
X˜ and let d be the size of the orbit [z]. Define a sign τ[z] to be +1 if ϕd fixes z1 and z2 ,
while τ[z] = −1 if ϕd switches z1 and z2 .
Proposition 4.2. Let nk = |Σk | be the number of singular k-points of X and let sk = |Ck |
be the number of irreducible k-components of X˜ . Then
Y
det −ϕ|H(T )⊗Z Q = (−1)nk +sk +1
τ[z] .
[z]∈Σk
Proof. We tensor (4.3) with Q over Z and compute det(−ϕ) on each term in the exact
sequence, beginning on the right side, where ϕ acts trivially on Z. Hence det (−ϕ|Q ) = −1.
Consider the orbit O of a vertex of Γ under ϕ. Let S be the Gal(k/k)-invariant submodule of C0 (Γ, Z) generated by the elements of O and let d = |O|. Then ϕ acts on O
as a cyclic permutation of length d and sign (−1)d−1 , so we have
det(−ϕ|S⊗Z Q ) = (−1)d sign ϕ = (−1)d (−1)d−1 = −1.
12
Note that each such orbit corresponds to a component of X˜ defined over k. Hence the
number of distinct orbits is sk . By decomposing C0 (Γ, Z) according to these distinct orbits
we find that
det −ϕ|C0 (Γ,Z)⊗Z Q = (−1)sk .
Consider the orbit O1 = [z] of a node z of X , z ∈ Σ, under the action of ϕ. The
Galois conjugates of the associated edge ez on the graph Γ generate a Gal(k/k)-invariant
submodule S1 of C1 (Γ, Z) whose rank is d1 = |O1 | and on which ϕd1 acts as multiplication
by τ[z] . It follows as above that
det (−ϕ|S1 ⊗Z Q ) = −τ[z] .
Now each such orbit [z] corresponds to an element of Σk . Since there are nk distinct
orbits, we have
Y
det −ϕ|C1 (Γ,Z)⊗Z Q = (−1)nk
τ[z] .
[z]∈Σk
Corollary 4.3. Let k 0 ⊂ k be the quadratic extension of k and let nk0 = | Σk0 | be the
number of singular k 0 -points of X . Then
det −ϕ|H(T ) = (−1)nk0 +nk +sk +1 .
Proof. For each singular point [z] of X defined over k, there are one or two singular points
defined over k 0 according as τ[z] = −1 or +1. Hence
Y
τ[z] = (−1)nk0 .
[z]∈Σk
5. Examples
In this section we give examples of hyperelliptic curves C of genus 2 over Q for which
we can compute root numbers W of their Jacobians J(C) using our results in the previous
sections. More precisely, each Jacobian in the examples has additive reduction at p = 3
(i.e., in the above notation K = Q3 and in (2.2) the connected component A0vK of the
reduction of J(C) at p = 3 equals U) and has totally toroidal semistable reduction over a
wildly ramified extension of Q3 satisfying conditions of Proposition 3.2 (i.e., in the above
notation A0vF is a torus). Thus in the notation of Section 2.2 above for K = Q3 we have
ρav is trivial, W (ρav ) = 1, V IK ∼
= (Tl (A0vK ) ⊗Zl Ql ) ⊗ı C is trivial (since A0vK is unipotent),
0
so that δ(ρ ) = 1, and hence by (2.7) and Proposition 2.2
(5.1)
W3 = W (J(C)3 ) = (det ρ¯tor )(−1),
which can be computed using Proposition 3.2. At other primes q 6= 3 each C (and hence
J(C) by Example 8 on p. 246 in [BLR90]) has stable reduction over Qq , so that
Wq = W (J(C)q ) = δ(ρ0 ),
13
which can be computed using Proposition 4.2.
Let
f (x) = x3 + 3ax2 + 3bx + 3c,
(5.2)
a, b, c ∈ Z, 3 - c.
Let Q be a fixed algebraic closure of Q. Let L ⊂ Q be the splitting field of f over Q and
factor f (x) = (x − r1 )(x − r2 )(x − r3 ), r1 , r2 , r3 ∈ L. Since f is Eisenstein at 3, the inertia
subgroup Iv (L/Q) of Gal(L/Q) at a place v over 3 in L contains an element σ of order
3. Furthermore, Iv (L/Q) is isomorphic to the cyclic group C3 if n is even and Iv (L/Q)
is isomorphic to the symmetric group S3 if n is odd.
Thanks to the action of σ, the various differences
of roots of f have the same v-adic
Q
valuation, say m = v(ri − rj ) for all i 6= j. But 1≤i<j≤3 (ri − rj )2 = disc(f ), so
(
1
1
n,
n is odd,
m = v(disc(f )) = v(3) n =
6
6
n/2, n is even.
Fix an element t in the ring of integers OL such that v(t) = 1 and write
r2 − r1 = tm u and r3 − r1 = tm u0 .
Then u and u0 embed as v-adic units in the completion Lv . Furthermore, u and u0 are
distinct modulo t, since tm (u0 − u) = r3 − r2 also has v-adic valuation m. By stretching
and translation, we have
(5.3)
f (tm X + r1 ) = tm X(tm X − tm u)(tm X − tm u0 ) = t3m X(X − u)(X − u0 ).
Note that
disc(f ) = −27(4b3 + 12a3 c + 9c2 − 18cab − 3a2 b2 ).
Hence, we have


3,
n = 4,

5,
P
Proposition 5.1. Let g(x) = 5j=0 cj xj
9 | c0
if 3 - b,
if 3 | b and 3 - a,
if 3 | b and 3 | a.
∈ Z[x] with
and
3 | c1 , c2 , c3 .
Given f (x) as in (5.2), assume that the polynomial f (x)2 + 3n−1 g(x) has distinct roots in
Q. Then the Jacobian of a hyperelliptic genus 2 curve C over Q defined via
C : y 2 = f (x)2 + 3n−1 g(x)
has additive reduction over Q3 , acquires totally toroidal reduction over Lv , and Lv is the
minimal Galois extension of Q3 over which it acquires semistable reduction.
14
Proof. By (5.3), the change of variables x = tm X + r1 , y = t3m Y leads to a model C 0 of
C over Lv of the form
(5.4)
C 0 : Y 2 = X 2 (X − u)2 (X − u0 )2 + tG(X),
in which the coefficients of G(X) = 3n−1 g(tm X + r1 )/t6m+1 are v-integral because of the
congruences imposed on the coefficients of g(x). Indeed, let
(
2, n is odd,
e=
1, n is even.
Then v(3) = 3e and v(tm ) = m > v(r1 ) = e. It is enough to choose αj = ord3 (cj ), so that
t6m+1 | 3n−1 cj (r1 )j for all j. Equivalently,
6m + 1 − je
− n + 1,
3e
which leads to the congruences stated in the proposition.
Clearly, C 0 defines a stable curve (see Definition 1.1 on p. 76 in [DM69]) over the
ring of integers OLv of Lv , whose reduction C˜ over the residue field k of OLv consists of
two projective lines intersecting transversally at three points. Let J denote the Jacobian
variety of C considered as an abelian variety over Lv . Then the connected component
Y of the reduction of J over k (considered over an algebraic closure k of k by extension
˜
of scalars) is isomorphic to Pic0 (C/k)
(by the results in [R70]) and hence J has totally
toroidal reduction over k. Indeed, let α denote the dimension of the abelian variety part
of Y and let β denote the dimension of the toric part of Y. Then α equals the sum of
genera of the irreducible components of C˜ and β equals the first Betti number of C˜ (see
Example 8 on p. 246 and Corollary 12 on p. 250 in [BLR90]).
Let v denote the 3-adic valuation on Q with v(3) = 1. One can easily check that C has
a model over Z of the form
αj ≥
y 2 = (x3 + e1 x2 + e2 x + e3 )2 + e4 x2 + e5 x + e6 ,
∀ei ∈ Z,
where v(e3 ) = 1, v(ei ) ≥ 1, i = 1, 2, v(ei ) ≥ 2, i = 4, 5, 6. It follows from the results
on p. 151–152 in [L94] that 3 divides the degree of the minimal Galois extension of Q3
over which C has stable reduction. Furthermore, the closed fiber F of a minimal model
of C over Z3 has type [IIIN ] for a natural number N (see [NU73]). In particular, F has
no cycles and its irreducible components are projective lines. As was explained at the
end of the previous paragraph, this implies that the reduction of J(C) over Q3 must be
additive. Finally, by the Deligne-Mumford theorem on stable curves and their Jacobians
(Theorem 2.4 on p. 89 in [DM69]), Lv is the minimal Galois extension over which the
Jacobian J(C) of C has semistable reduction.
We now give a hyperelliptic curve C of genus 2
C : y 2 + Q(x)y = P (x),
P, Q ∈ Z[x],
15
where Q(x) is monic of degree 3 and deg P (x) ≤ 5. The discriminant ∆ of C is calculated
as follows
∆ = 2−12 disc(F ), where F = Q2 + 4P
([L96], p. 4581). We give a cubic f , Eisenstein at 3 as in (5.2), such that F = f 2 + 3n−1 g
with g as in Proposition 5.1. By Proposition 5.1 this is enough to guarantee that the
Jacobian J = J(C) of C acquires totally toroidal semistable reduction over the completion
Lv of the splitting field L of f at each place v over 3. Moreover, using Proposition 5.1,
Equation (5.1), and Proposition 3.2, one could easily check that W3 = 1 if Iv (L/Q) ∼
= C3
∼
and W3 = −1 if Iv (L/Q) = S3 .
In the examples, for a prime q 6= 3 dividing ∆, we have ordq (∆) = 1. Hence the
equation y 2 = F (x) defines a minimal Weierstrass model of C over Zq (see [L96]) and the
reduction C of C at q has the form
C : y 2 ≡ (x − r)2 h(x) mod q,
h(x) ∈ Z[x],
h(r) 6= 0 mod q,
deg h = 4.
There is one ordinary double point corresponding to x = r and the blow-up of C at x = r
is a curve of genus 1. This implies that J has semistable reduction at q = 3 whose abelian
variety and toric parts are both of dimension one. Since C is irreducible, we get sq = 1
and nq = 1 as well. The root number therefore is
Wq = −τ[z] = −legendre(h(r), q),
so that Wq = −1 if the slopes above are rational and Wq = 1 otherwise.
Case 1: Iv (L/Q) ∼
= C3 , f = x3 + 3x2 − 3, n = 4
Q
x3 + x2 + 3
5
P
−80x + 2x4 − 3x3 − 6x2
∆
325 · 13 · 109 · 4507
g
−12x5
W =1
W3 = W109 = 1
W13 = W4507 = −1
= C3 , f = x3 + 3x2 − 3, n = 4
Q
x 3 + x2 + 3
P
82x5 + 2x4 − 3x3 − 6x2
∆
325 · 13 · 561359
g
12x5
W = −1
W3 = W13 = 1
W561359 = −1
16
= C3 , f = x3 + 3x2 − 3, n = 4
Q
(x + 1)3
P
12x4 + 34x3 + 12x2 + 39x + 2
∆
321 · 852263
4
g
2x + 6x3 + 3x2 + 6x
W = −1
W3 = 1
W852263 = −1
= S3 , f = x3 − 3x − 3, n = 3
Q
x3 + x + 1
4
P
−2x − 2x3 + 2x2 − 77x + 2
∆
319 · 383 · 1607
g
−36x
W =1
W1607 = 1
W3 = W383 = −1
= S3 , f = x3 − 3x − 3, n = 3
Q
x3
4
3
P
−6x − 15x − 18x2 − 9x − 18
∆
316 · 23 · 37 · 167
g
−2x4 − 6x3 − 9x2 − 6x − 9
W =1
W23 = W167 = 1
W3 = W37 = −1
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