ROOT NUMBERS OF ABELIAN VARIETIES AND - Penn Math

98 Pages·2005·0.39 MB·English

ROOT NUMBERS OF ABELIAN VARIETIES AND
REPRESENTATIONS OF THE WEIL-DELIGNE GROUP
Maria Sabitova
A Dissertation
in
Mathematics
Presented to the Faculties of the University of Pennsylvania in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
2005
Ted Chinburg
Supervisor of Dissertation
David Harbater
Graduate Group Chairperson
Acknowledgments
I would like to thank my advisor Ted Chinburg for suggesting the problem and useful
discussions. I am also grateful to Siegfried Bosch, Ching-Li Chai, Robert Kottwitz,
and Michel Raynaud for answering my questions related to the uniformization theory,
and especially to Michel Raynaud for providing the reference [Ra].
Thanks to everybody in the math department who has supported me, especially
Ted Chinburg, Alexander Kirillov, Tony Pantev, and Janet Burns.
Last, but not least, I would like to express my deepest gratitude to my parents
and Mitya Boyarchenko.
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ABSTRACT
ROOT NUMBERS OF ABELIAN VARIETIES AND REPRESENTATIONS OF
THE WEIL-DELIGNE GROUP
Maria Sabitova
Ted Chinburg
We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves
over the field of rational numbers. Our result applies to abelian varieties over number
fields. Namely,undercertainconditionswhichnaturallyextendtheconditionsusedby
D.Rohrlich, weshowthattherootnumberW(A,τ)associatedtoanabelianvarietyA
over a number field F and a complex finite-dimensional irreducible representation τ of
Gal(F/F) with real-valued character is equal to 1. In the case where the ground field
is Q, we show that our result is consistent with a refined version of the conjecture
of Birch and Swinnerton-Dyer. We also give a description of unitary, orthogonal,
and symplectic admissible representations of the Weil-Deligne group of a local non-
Archimedean field.
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Contents
1 Introduction 1
2 Root numbers of abelian varieties over local non-Archimedean fields
of characteristic zero 10
2.1 General facts and notation . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Case of an abelian variety with potential good reduction. . . . . . . . 15
2.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Root numbers of abelian varieties over number fields (Theorem A) 31
3.1 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Special cases of Theorem A . . . . . . . . . . . . . . . . . . . . . . . 35
4 Representations of the Weil-Deligne group (Theorem B) 43
4.1 Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 An application of Theorem B . . . . . . . . . . . . . . . . . . . . . . 49
A 57
iv
B 72
C 77
D 81
v
Chapter 1
Introduction
One of the main objects of study in this thesis is the root number W(A,τ) associ-
ated to an abelian variety A of dimension g over a number field F and a continuous
irreducible complex finite-dimensional representation τ of Gal(F/F) with real-valued
character. The root number W(A,τ) is a complex number of absolute value 1. As-
sume for simplicity that F = Q. Then W(A,τ) appears in the following conjectural
functional equation:
Λ(A,τ,s) = W(A,τ)·Λ(A,τ∗,2−s), (1.0.1)
where s ∈ C, τ∗ is the contragredient of τ, and
Λ(A,τ,s) = Cs ·Γ(s)gdimτ ·L(A,τ,s)
for some positive constant C and the twisted L-function L(A,τ,s) which is a mero-
morphic function of s defined in a right half-plane. This function is conjectured to
have an analytic continuation to the entire complex plane. Since τ has real-valued
1
character, τ ∼= τ∗. Assuming (1.0.1) and considering the power series expansion of
L(A,τ,s) about s = 1, we get:
W(A,τ) = (−1)ords=1L(A,τ,s). (1.0.2)
In this thesis we generalize a result by D. Rohrlich for elliptic curves ([Ro2], p.
313, Prop. E) to abelian varieties. We prove the following theorem:
Theorem 1.0.1. Let F be a number field, L a finite Galois extension of F, and
τ an irreducible complex finite-dimensional representation of Gal(L/F) with real-
valued character. Let g be a fixed positive integer and assume that the decomposition
subgroups of Gal(L/F) at all the places of F lying over all the primes less or equal to
2g+1 are abelian. If the Schur index m (τ) is 2 then W(A,τ) = 1 for every abelian
Q
variety A of dimension g over F.
If F = Q then Theorem 1.0.1 is predicted by the conjectures of Birch-Swinnerton-
Dyer and Deligne-Gross. Namely, the conjectures of Birch-Swinnerton-Dyer and
Deligne-Gross imply
ord L(A,τ,s) = hσ ,τi, (1.0.3)
s=1 A
where σ is the natural representation of Gal(Q/Q) on C⊗ A(Q) and hσ ,τi is the
A Z A
multiplicity of τ in σ ([Ro3], p. 127, Prop. 2). Thus, we get from (1.0.2) and (1.0.3):
A
W(A,τ) = (−1)hσA,τi.
Since σ is realizable over Q and τ is irreducible, m (τ) divides hσ ,τi. Thus, if
A Q A
m (τ) = 2 then W(A,τ) = 1 for every abelian variety A over Q if (1.0.3) is true (cf.
Q
2
[Ro2], p. 313).
To prove Theorem 1.0.1 we use the following formula:
Y
W(A,τ) = W(A ,τ ),
v v
v
wherev runsthroughalltheplacesofF,A = A× F ,F denotesthecompletionofF
v F v v
with respect to v, and τ is the restriction of τ to Gal(F /F ) ,→ Gal(F/F). To define
v v v
W(A ,τ ) for every place v let σ0 denote the representation of the Weil-Deligne group
v v v
W0(F /F ) associated to the first cohomology of A . Then W(A ,τ ) = W(σ0 ⊗τ ),
v v v v v v v
where τ is viewed as a representation of W0(F /F ). We will in fact show the
v v v
following stronger result:
Theorem 1.0.2 (Theorem A). W(A ,τ ) = 1 for all v under the hypotheses of
v v
Theorem 1.0.1.
First, we describe W(A ,τ ) when τ is a complex finite-dimensional continuous
v v v
representation of Gal(F /F ) with real-valued character. If v is an infinite place then
v v
σ0 is associated to the components of H1(A (C),C) in the Hodge decomposition. We
v v
show in Lemma 3.1.1 that
W(A ,τ ) = (−1)gdimτv. (1.0.4)
v v
If v is a finite place, then
(cid:15)(σ0 ⊗τ ,ψ ,dx )
W(σ0 ⊗τ ) = v v v v ,
v v |(cid:15)(σ0 ⊗τ ,ψ ,dx )|
v v v v
where ψ is a nontrivial additive character of F and dx is a Haar measure on F .
v v v v
Here σ0 is isomorphic to the representation of W0(F /F ) afforded by H1(A ), where
v v v l v
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l is a rational prime different from the residual characteristic of F . It is known that
v
Hl1(Av) ∼= Vl(Av)∗ as Gal(Fv/Fv)-modules over Ql, where Vl(Av) = Tl(Av) ⊗Zl Ql,
T (A ) is the l-adic Tate module of A , and V (A )∗ denotes the contragredient of
l v v l v
V (A ). Thus, wecan assume thatσ0 is therepresentation ofW0(F /F )associated to
l v v v v
V (A )∗. Clearly, W(σ0⊗τ )doesnotdependonthechoiceofdx anditturnsoutthat
l v v v v
W(σ0 ⊗τ ) does not depend on the choice of ψ either. Moreover, W(σ0 ⊗τ ) = ±1
v v v v v
(see Section 2.1).
We consider two cases: A is an abelian variety with potential good reduction
v
and the general case. If A has potential good reduction, it follows from N´eron-Ogg-
v
Sˇhafareviˇc criterion that σ0 is actually a representation of the Weil group W(F /F ).
v v v
If the characteristic of the residue class field k of F is greater than 2g+1, we use the
v v
theory of Serre-Tate together with methods of the representation theory to describe
theclassofσ0⊗ω1/2 intheGrothendieckgroupofvirtualrepresentationsofW(F /F )
v v v v
(Corollary 2.2.7, Formula (2.2.4)). Here ω is the one-dimensional representation of
v
W(F /F ) given by
v v
ω | = 1, ω (Φ ) = q−1,
v Iv v v v
where I is the inertia subgroup of Gal(F /F ), Φ is an inverse Frobenius element
v v v v
of Gal(F /F ), and q = card(k ). Since the root number of representations of
v v v v
W(F /F ) is multiplicative in short exact sequences, this result enables us to prove
v v
the following formula for W(σ0 ⊗τ ) when char(k ) > 2g+1 (cf. Proposition 2.2.9):
v v v
W(σ0 ⊗τ ) = detτ (−1)l1 ·βdimτv ·γl2 ·(−1)hνv,τvi, (1.0.5)
v v v
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where l ∈ Z, β = ±1, γ = ±1, l = h1,τ i+hη ,τ i, η is the unramified quadratic
1 2 v v v v
character of F×, and ν is a representation of Gal(F /F ) realizable over Q (cf. [Ro2],
v v v v
p. 318, Thm. 1).
In the general case we use the theory of uniformization of abelian varieties. Ac-
cording to this theory there exists a semi-abelian variety G over F and a discrete
v v
subgroup Y of G such that, in terms of rigid geometry, A is isomorphic to the
v v v
quotient G /Y . The semi-abelian variety G fits into an exact sequence
v v v
0 −→ T −→ G −f→v B −→ 0, (1.0.6)
v v v
where B is an abelian variety over F with potential good reduction, T is a torus
v v v
over F of dimension r; Y is an ´etale sheaf of free abelian groups over Spec(F ) of
v v v
rank r. To describe σ0 in this case we use a formula of M. Raynaud ([Ra], p. 314)
v
which gives the action of the inertia group I on the ln-torsion points of an abelian
v
variety over a non-Archimedean local field in the case when the uniformization data
splits. We need this formula to show that in this case
σ0 ∼= κ ⊕(χ ⊗ω−1 ⊗sp(2)), (1.0.7)
v v v v
where κ is the representation of W0(F /F ) associated to the natural l-adic repre-
v v v
sentation of Gal(F /F ) on V (B )∗,
v v l v
χ : Gal(F /F ) −→ GL (Z)
v v v r
is the representation of Gal(F /F ) corresponding to the Galois module Y (F ), and
v v v v
sp(2) is given by (2.1.1) (see Proposition 2.3.1). Since the root number of a direct
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