Modularity of Shafarevich-Tate Groups

William Stein

http://modular.fas.harvard.edu

May 22, 2004

Goal

The goal of this 40-minute talk is to explain the meaning of the following conjecture and give evidence for it.

Conjecture (–). If A is a modular abelian variety, then the Shafarevich-Tate group � (A) is modular.

Table of Contents

1. Elliptic Curves and Modular Abelian Varieties 2. The Birch and Swinnerton-Dyer Conjecture 3. Visibility of Shafarevich-Tate groups

4. Modularity of Shafarevich-Tate groups

5. Some Data

1. Elliptic Curves and Abelian Varieties

Elliptic curve over Q: y^{2} = x^{3} + ax + b with a, b ∈ Q and Δ = −16(4a^{3} +27b^{2}) ≠ 0

Abelian variety: Any complete group variety.

Examples: Jacobians of curves. Elliptic curves are the abelian varieties of dimension one. Modular abelian varieties.

Modular Curves, Modular Forms

Congruence Subgroup:

Γ_{0}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z) such that N | c\} .

Modular Curve: X_{0}(N) = Γ_{0}(N) \ (upper half plane) ∪ (cusps) Example:X_{0}(39)

Algebraic structure over Q:

y^{2} = (x^{4} − 7x^{3} + 11x^{2} − 7x + 1)(x^{4} + x^{3} − x^{2} + x + 1).

Modular Abelian Varieties

Modular Jacobian: J_{0}(N) = Jac (X_{0}(N)) Modular Abelian Variety: Any abelian variety quotient of J_{0}(N) (or of J_{1}(N), where J_{1}(N) is defined using Γ_{1}(N)). Theorem (Wiles et al.): All elliptic curves over Q are modular. Cusp Forms: S_{2}(Γ_{0}(N)) ≅ H^{0}(X_{0}(N)_{C}, Ω^{1}) Hecke Algebra: T = Z[T_{1}, T_{2}, T_{3}, …] ⊂ End (S_{2}(Γ_{0}(N))) Shimura: T-eigenform f ∈ S_{2}(Γ_{0}(N)) gives A_{f} = J_{0}(N)/I_{f}J_{0}(N), where I_{f} = Ann_{T}(f). We have dim A_{f} = [Q(a_{2}(f), …) : Q].

2. The Birch and Swinnerton-Dyer Conjecture

Let A be an abelian variety over Q (e.g., A = A_{f} modular).

Conjecture of :

1. r = rank A(Q) =^{conj.} ord_{s=1} L(A, s), and

2. \frac{L^{(r)}(A, 1)}{r!} = \frac{∏ c_{p} · Ω_{A} · Reg (A) · #� (A)}{#A(Q)_{tor} · #A^{∨}(Q)_{tor}} .

Results: Kolyvagin, Kato, Rubin, etc.

Shafarevich-Tate Group

Definition:

� (A) = Ker (H^{1}(K, A) → _{all v}^{⨁} H^{1}(K_{v}, A)) .

H^{1}(K, A) is Galois cohomology. Interpret geometrically as the Weil-Chatalet group:

WC (A/K) = { principal homogenous spaces X for A }/ ∼ . � (A) is the subgroup of locally trivial classes. Example:

3x^{3} + 4y^{3} + 5z^{3} = 0 ∈ � (x^{3} + y^{3} + 60z^{3} = 0)[3].

3. Visibility of Sha

0 → A ⟶^{i} B → C → 0

1998 – Barry Mazur introduced Visibility:

Vis_{B}(H^{1}(K, A)) = Ker (H^{1}(K, A) → H^{1}(K, B))
≅ Coker (B(K) → C(K))
Vis_{B}(� (A)) = Ker (� (A) → � (B)).

Visibility

[figure: X = π^{−1}(P); P; c; 0; A(K); B(K); π; C(K); Vis_{B}(H^{1}(K, A)); 0]

Why? To write down X using equations is terrifying; to give P is just to give a rational point. Visibility concisely encodes connections between Mordell-Weil and Shafarevich-Tate groups.

Give nonzero c ∈ � (A)[5] with dim (A) = 20 by giving

(0, 0) ∈ [y^{2} + y = x^{3} + x^{2} − 2x]

Everything is Visible Somewhere

Theorem (–): If c ∈ H^{1}(K, A) then there exists B such that i : A ↪ B and c ∈ Vis_{B}(H^{1}(K, A)). Proof. Let L be such that res_{L/K}(c) = 0. Then

Note: If A/Q is modular and L is abelian, then B is modular.

[figure: c; res_{L/K}(c) = 0; H^{1}(K, A); H^{1}(K, Res _{L/K}(A_{L})); H^{1}(L, A)]

4. Modularity of Sha

Definition (Modular): An element c ∈ � (A) is modular if it is visible in a modular abelian variety. I.e., if there is a factor B of some J_{0}(N) (or J_{1}(N)) and an inclusion i : A ↪ B such that i_{∗}(c) = 0. Torsor corresponding to c is modular in “usual” sense.

Modularity Conjecture (–):

If A is modular, then every element of � (A) is modular.

Theorem (Klenke, Mazur, Stein): If c ∈ � (E)[p] with p = 2, 3 and E an elliptic curve, then c is modular.

Related questions:

1. Levels N such that such that c is modular of level N?

2. Which genus one curves are modular?

Why Care about Modularity?

• It could yield results about structure of � (A), e.g., finiteness.

• It would give a nice “explanation” of where all � (A) “comes from” — it all comes from Mordell-Weil groups.

• It motivates proving new results about the arithmetic of modular abelian varieties.

• It provides powerful computational tools for explicitly working with Shafarevich-Tate groups.

Visibility Construction

Theorem (Agashe, –): Suppose A, B ⊂ J_{0}(N), that B[p] ⊂ A, and other technical hypotheses. Then

B(Q)/pB(Q) ↪ Vis_{J_{0}(N)}(� (A)).

Proof. Use the following diagram, chase some exact sequences, and apply subtle properties of Néron models. (Also more general Hecke-equivariant version, proved recently with Jetchev.)

\xymatrix{B[p] \ar[r] \ar[d] & B \ar[r]^{p} \ar[d] & B \ar[d] \\ A \ar[r] & J_{0}(N) \ar[r] & C.}

5. Some Data

Suppose A ⊂ J_{0}(N)

(a) Visibility of � (A) in J_{0}(N), when A is an elliptic curve.

(b) Visibility of � (A) in J_{0}(N), general A.

(c) Visibility of � (A) in J_{0}(Np), general A. (Modularity)

(a) Elliptic Curve tables from Cremona-Mazur

Visibility of � (A) in J_{0}(N), when A = E is an elliptic curve.

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(b) Abelian Variety Tables from Agashe-Stein

Visibility of � (A) in J_{0}(N), general A.

(c) Visibility at Higher Level –

Evidence for Modularity Conjecture

Visibility of � (A) in J_{0}(Np), general A.

Recall Ribet’s theorem...

Ribet Level Raising

Suppose • f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) a newform

• λ ⊂ Z[a_{1}, a_{2}, …] a nonzero prime ideal s.t. A_{f}[λ] irreducible. Theorem: ap + p + 1 ≡ 0 (mod λ) ⟹ there exists a p-newform g ∈ S_{2}(Γ_{0}(Np)) such that

• i(A_{f}[λ]) = A_{g}[λ] some i : J_{0}(N) → J_{0}(Np), and

• sign of functional equations for L(f, s) and L(g, s) same.

Big Computation: For every level N up to 5000 (and more), use my modular forms package in MAGMA to provably compute:

1. Each newform f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)).

2. Whether or not L(f, 1) = 0.

3. Whether or not ord_{s=1} L(f, s) is even.

4. Characteristic polynomials of a_{2}, a_{3}, a_{5}, …, a_{19} . (I hope to redo this computation using only open-source software that I’m currently writing.)

Probable Modularity

• Two forms f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n} are probably congruent mod ℓ (away from level) if for p < 20 with p ∤ N_{f}N_{g} we have

ℓ | resultant (charpoly (a_{p}), charpoly (b_{p})).

• If A = A_{g} ⊂ J_{0}(N), then there is probably a nonzero element in � (A)[ℓ] modular of level Np if there is f of level Np such that:

1. f and g are probably congruent modulo ℓ, and

2. ord_{s=1} L(f, s) is positive and even.

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Questions

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