Visibility of Shafarevich-Tate groups of modular abelian varieties

William A. Stein^{∗}

July 2000

Abstract These are the notes for my July 2000 Durham talk on visibility of Shafarevich-Tate groups. I timed them in quickly the night before my talk, in order to make precise what I would say.

1 Modular abelian varieties

A central class of abelian varieties of interest in number theory are the modular abelian varieties of GL_{2}-type. It is now known that every elliptic curve over Q is a member of this class. Such abelian varieties are constructed as follows. Let f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N), C) be a newform. The Hecke algebra T = Z[… T_{n} …] acts on f, and there is a surjective homomorphism T → O_{f} = Z[… a_{n} …] sending T_{n} to a_{n}. The kernel I_{f} of this homomorphism is the annihilator of f in T. Attached to I_{f} there is a quotient A_{f} = J_{0}(N)/I_{f}(J_{0}(N)), where J_{0}(N) is the Jacobian of the modular curve X_{0}(N). The abelian variety A_{f} has dimension d = [Q(… a_{n} …) : Q], and the dual A_{f}^{∨} is an abelian subvariety of J_{0}(N).

There are two groups attached to an abelian variety A over Q. One is the Mordell-Weil group A(Q), which is a finitely generated abelian group, and the other is the Shafarevich-Tate group � (A) = ker(H^{1}(Q, A) → ∏_{v} H^{1}(Q_{v}, A)), which is (conjecturally!) a finite abelian group. This talk is about a link between these two groups, which is frequently signaled by the term “visibility”. Before proceeding further, we state a fundamental conjecture that both guides and stimulates our work.

Conjecture 1.1 (Birch and Swinnerton-Dyer). Let A be an abelian variety over Q. Then � (A) is finite, and the following formula holds:

L(A, 1) = \frac{∏ c_{p} · Ω_{A}}{#A(Q) · #A^{∨}(Q)} · #� (A),

where the right hand side should be interpreted as 0 if A(Q) is infinite. (There is a refinement when L(A, 1) = 0.)

∗Harvard University, was@math.harvard.edu

2 Mazur: Visualize Sha!

Let A be an abelian variety over Q. To give an elements of � (A) is the same as giving an algebraic variety X over Q equipped with an action A × X → X satisfying analogues of the usual axioms for a simply transitive group action. Where can we find these varieties X? As translates of A inside a bigger abelian variety J!

Fix an embedding of A into an abelian variety J, and let C be the quotient:

A ↪ J → → C.

Taking cohomology, reveals the following diagram:

\xymatrix{ &  &  & Vis_{J}(� (A)) \ar@{^{(}->}[r] & � (A) \ar@{^{(}->}[d] \ar[r] & � (J) \ar@{^{(}->}[d] \\ 0 \ar[r] & A(Q) \ar[r] & J(Q) \ar[r] & C(Q) \ar[r] & H^{1}(Q, A) \ar[r] & H^{1}(Q, J)}

Definition 2.1 (Visible part). The visible part of � (A) with respect to the embedding A ↪ J is the subgroup

Vis_{J}(A) = ker(� (A) → � (J)).

Likewise, the visible part of H^{1}(Q, A) is

Vis_{J}(H^{1}(Q, A)) = ker(H^{1}(Q, A) → H^{1}(Q, J)).

Question 2.2 (Mazur). Is every element c ∈ H^{1}(Q, A) visible somewhere?

Johan de Jong answered this question in the affirmative, when dim A = 1; however, his proof, which exploits Azamuya algebras over Spec(Z), is not elementary. Fortunately, Mazur’s question has an elementary answer.

Proposition 2.3 (Folklore?). Given c ∈ H^{1}(Q, A) there is an abelian variety J and an embedding ι : A ↪ J such that ι_{∗}(c) = 0 ∈ H^{1}(Q, J).

Proof. Because cochains are continuous, there is a finite extension K of Q such that res_{K}(c) = 0 ∈ H^{1}(K, A). The Shapiro lemma implies that H^{1}(K, A) is canonically isomorphic to H^{1}(Q, Res_{K/Q} A_{K}), where Res_{K/Q} A_{K} is the Weil restriction of scalars of A_{K}. Furthermore, there is a canonical embedding ι : A ↪ J = Res_{K/Q} A_{K}, and this embedding sends c to res_{K}(c) = 0.

Corollary 2.4. Every c ∈ � (E)[2] is visible in an abelian surface, and every c ∈ � (E)[3] is visible in a three-dimensional abelian variety.

Using more sophisticated techniques, Mazur proved that every c ∈ � (E)[3] is visible in an abelian surface.

For the rest of this talk, we return to the case where A is a attached to a modular form.

Question 2.5 (Mazur). Let f be a newform of level N. Can one visualize the elements c ∈ � (A_{f}^{∨}) inside J_{0}(N)? Inside J_{0}(M) for some multiple M of N?

3 Some data

Mazur, Cremona, and Logan collected data on visibility of Sha using only elliptic curves and congruences between elliptic curves. The data is impressive, but probably only because � is very small in the range considered. As Frey emphasized in his talk, there are very few intersections between elliptic curves.

The author made a table of rational numbers L(A_{f}, 1)/Ω_{A_{f}} for all f ∈ S_{2}(Γ_{0}(p)), for p ≤ 2161. To analyze the resulting data, let us momentarily assume the truth of the Birch and Swinnerton-Dyer conjecture. We find 38 of the A_{f} have the property that an odd prime ℓ divides #� (A_{f}). Of these, the full odd part of � (A_{f}) is visible in 22 cases, whereas it is invisible in all of the remaining 16 cases. A small selection of data is recorded in the table below.

p  dim A_{f} \sqrt{#� (A_{f})^{odd}}  dim B
389  20  5 (VIS)  1
433  16  7 (VIS)  1
1061  46  151 (VIS)  2
1091  62  7 (INV, so far)  —
1429  64  5 (VIS at level 2 · 1429) 2
2111  112  211 (INV, so far)  —
2333  101  83341 (VIS)  4
2849  1  3 (VIS at level 3 · 2849) 1
5389  1  3 (VIS at level 7 · 3849) 1

4 Criterion for visibility

Theorem 4.1 (Stein). Let A, B ⊂ J, and let N be divisible exactly by the primes of bad reduction for J. Suppose that

• (A ∩ B)(\overline{Q}) is finite,

• A(Q) is finite,

• B has purely toric reduction at each ℓ | N.

Let p be a prime such that

1.

p ∤ 2N · #(\frac{J(Q)}{A(Q)})_{tor} · B(Q)_{tor} · ∏_{ℓ|N} #Φ_{A,ℓ}(\overline{F}) · #Φ_{B,ℓ}(F_{ℓ}),

2. B[p] ⊂ A (this condition can be relaxed).

Then

B(Q)/pB(Q) ↪ Vis_{J}(� (A)).

Sketch. Consider the diagram

\xymatrix{ & A ∩ B \ar[r] \ar[d] & B \ar[d] \ar[r]^{p} & B \ar[d] \\ 0 \ar[r] & A \ar[r] & J \ar[r] & C \ar[r] & 0.}

Taking cohomology, we arrive at the following key diagram:

\xymatrix{0 \ar[r] & B(Q) \ar[d] \ar[r]^{p} & B(Q) \ar[d] \ar[r] & B(Q)/pB(Q) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & J(Q)/A(Q) \ar[r] & C(Q) \ar[r] & Vis_{J}(H^{1}(Q, A)) \ar[r] & 0.}

Upon applying the snake lemma and using the above hypothesis, we deduce that B(Q)/pB(Q) ↪ Vis_{J}(H^{1}(Q, A)). The deduction that B(Q)/pB(Q) ↪ Vis_{J}(� (A)) involves a local analysis at each prime ℓ of Q.

5 Visibility at higher level

There is a 53-dimensional abelian subvariety A of J_{0}(1429) such that 5 divides the conjectural order of � (A). However, this part of � can not be visible in J_{0}(1429). Ribet’s level raising theorem supplies us with an infinite collection of primes p such that an image of A in J_{0}(pN) meets an abelian variety B in some 𝔪-torsion, where 𝔪 is an appropriate maximal ideal of residue characteristic 5 in T. For each of these B, probably (I haven’t proved this)

L(B, 1) ≡ L(A, 1) = 0 (mod 𝔪),

so either 5 | #� (B) or L(B, 1) = 0. In the latter case, the Birch and Swinnerton-Dyer conjecture predicts that B(Q) is infinite; the theorem above then implies that

5 | Vis_{J_{0}(pN)}(� (A)).

There are many similar examples, some of which have dim B = 1, and hence don’t rely on any conjectures. Based on this evidence, we make the following conjecture.

Conjecture 5.1 (Stein). Let N be a prime and A = A_{f}^{∨} ⊂ J_{0}(N). If c ∈ � (A), then there is a prime p such that δ_{1}^{∗}(c) + δ_{p}^{∗}(c) = 0 ∈ � (J_{0}(pN)).

6 Speculation: constructing points?

Constructing elements of Vis_{J_{0}(N)}(� (A)) is equivalent to constructing points on B. This observation might lead to an approach to proving the implication L(B, 1) = 0 implies B(Q) is infinite. At present, mathematicians seem to have no good construction of points when L(B, 1) = L′(B, 1) = 0. However, we do have Kolyvagin’s Euler systems when L(A, 1) ≠ 0!