Nonvanishing Twists and Visible Shafarevich-Tate Groups

William Stein

              http://modular.fas.harvard.edu

October 29, 2001 at Boston University

Contents

1 Terminology 1 2 A Conjecture About Nonvanishing of Twists 2 3 Avoiding p-torsion 2 4 Visualizing Mordell-Weil in Rank 0 Sha 3 5 A Connection with the BSD Conjecture 4 6 A Nonsquare Shafarevich-Tate Group 4

Let E be an elliptic curve over ℚ and let p be a prime. One of my long-term goals is to find a rank 0 modular abelian varieties A that fits into a short exact sequence

0 → A → J → E → 0

in such a way that the exact sequence of Galois cohomology induces an isomorphism

E(ℚ)/pE(ℚ) ≅ ker (� (A)[p] → � (J)) := Vis_{J}(� (A)[p]).

I am searching for such A because I hope to relate the rank 0 Birch and Swinnerton-Dyer formula to the conjecture that L(E, 1) = 0 if and only if E(ℚ) is infinite. Acknowledgement: Stoll, Chinta, Rohrlich, Gross, Mazur, and Poonen.

1 Terminology

Let N_{E} denote the conductor of E, and for each prime ℓ | N_{E} let Φ_{E,ℓ} denote the component group of the Néron model of E at ℓ.

Definition 1.1. A prime p is rigid for E if

p ∤ 2 · N_{E} · ∏_{ℓ |N_{E}} #Φ_{E,ℓ}(\overline{𝔽}_{ℓ})

and the Galois representation ρ_{E,p} : G_{ℚ} → Aut(E[p]) is irreducible^{1}.

2 A Conjecture About Nonvanishing of Twists

Suppose p is rigid for E. For every ℓ ≡ 1 (mod p), let χ_{p,ℓ} : (ℤ/ℓℤ)^{∗} → → μ_{p} be a Dirichlet character of order p and conductor ℓ.

Conjecture 2.1 (–). For every rigid prime p, there exists a prime ℓ ∤ N_{E} with ℓ ≡ 1 (mod p) such that L(E, χ_{p,ℓ}, 1) ≠ 0 and a_{ℓ}(E) ≢ ℓ + 1 (mod p).

Evidence: This conjecture is true for every pair (E, p) with p < 50 rigid and E any of the first 20 rank-one optimal quotients of J_{0}(N) or either of the first 2 rank-two optimal quotients. Also, people such as David Rohrlich and Gautam Chinta have proved similar statements, and Hoffstein is rumored to be working on a very similar question. Jack Fearnley of Montreal proved something similar in his thesis in the case when L(E, 1) ≠ 0.

3 Avoiding p-torsion

Let p be a rigid prime for E, and suppose there is a prime ℓ with the properties predicted by Conjecture 2.1. Let K/ℚ be the abelian extension corresponding to χ_{p,ℓ} : (ℤ/ℓℤ)^{∗} → μ_{p}.

The restriction of scalars J = Res_{K/ℚ} E_{K} is an abelian variety over ℚ of dimension p; it represents the function S ↦ E_{K}(S_{K}). There is a natural injection E ↪ J and a natural surjection tr : J → → E with connected kernel. If we let A = ker(tr) then since the composite E → J → E is multiplication by [K : ℚ], the setup is as follows:

\xymatrix{E[p] \ar[r] \ar[d] & E \ar[r]^{[p]} \ar[d] & E \ar[d] \\ A \ar[r] & J \ar[r]^{tr} & E.}

Lemma 3.1. If a_{ℓ}(E) ≢ ℓ + 1 (mod p), then

E(ℚ_{ℓ}), J(ℚ_{ℓ}), (J/E)(ℚ_{ℓ}), and Φ_{A,ℓ}(𝔽_{ℓ})

all have trivial p-torsion.

1Mazur proved in 1978 that if p > 163 then ρ_{E,p} is irreducible.

Proof. The action of the Frobenius automorphism Frob_{ℓ} ∈ Gal(ℚ_{ℓ}^{ur}/ℚ_{ℓ}) on E(ℚ_{ℓ}^{ur})[p] has characteristic polynomial

x^{2} − a_{ℓ}(E)x + ℓ ∈ 𝔽_{p}[x].

Our hypothesis on a_{ℓ}(E) implies that this polynomial does not have +1 as a root, so E(ℚ_{ℓ})[p] = {0}. We next prove that J(ℚ_{ℓ})[p] = {0}. By definition,

J(ℚ_{ℓ}) = E_{K}(ℚ_{ℓ} ⊗_{ℚ} K) = E_{K}(K_{v}),

where K_{v} is the completion of K at the unique prime over ℓ. Also, because ℓ ∤ Np, we have E(\overline{ℚ}_{ℓ})[p] = E(ℚ_{ℓ}^{ur})[p]. If z ∈ E(K_{v})[p] then ℚ_{ℓ}(z) ⊂ K_{v} ∩ ℚ_{ℓ}^{ur}, so since K_{v} is totally ramified, ℚ_{ℓ}(z) = ℚ_{ℓ}, so z = 0. It follows that J(ℚ_{ℓ})[p] = {0}.

To see that (J/E)(ℚ_{ℓ})[p] = {0} note that

(J/E)(ℚ_{ℓ}) ⊂ (J/E)(K_{v}) ≈ ∏ E(K_{v})

since J_{K_{v}} is ∏ E_{K_{v}} and the diagonal embedding E_{K_{v}} → ∏ E_{K_{v}} splits.

Finally, consider Φ_{A,ℓ}. Let A denote the Néron model of A over ℤ_{ℓ}. By Lang’s Lemma, A(𝔽_{ℓ}) → → Φ_{A,ℓ}(𝔽_{ℓ}). Thus if Φ_{A,ℓ}(𝔽_{ℓ})[p] ≠ {0}, then A(𝔽_{ℓ})[p] ≠ {0}. Since p ≠ ℓ, Hensel’s lemma (and formal groups) imply that A(ℚ_{ℓ})[p] ≠ {0}, contrary to the fact that J(ℚ_{ℓ})[p] = {0}.

4 Visualizing Mordell-Weil in Rank 0 Sha

Let A ↪ J be a closed immersion of abelian varieties. Then Barry Mazur defined

Vis_{J}(� (A)) := ker(� (A) → � (J)).

Theorem 4.1. Suppose A, B ⊂ J and (A ∩ B)(\overline{ℚ}) is finite as a set. If p is a prime such that B[p] ⊂ A and

p ∤ 2 · N_{J} · #(J/B)(ℚ) · #B(ℚ)_{tor} · ∏_{ℓ|N_{J}} (#Φ_{A,ℓ}(𝔽_{ℓ}) · #Φ_{B,ℓ}(𝔽_{ℓ})) ,

then

B(ℚ)/pB(ℚ) ≅ Vis_{J}(� (A)[p]).

(Note: If (J/B)(ℚ) is infinite, then no primes satisfy the hypothesis of the theorem.)

Amod Agashe and I prove most of this theorem in our paper Visibility of Shafarevich-Tate Groups of Abelian Varieties. The proof uses the snake lemma and a careful local analysis at each prime.

Theorem 4.2. Let E be an elliptic curve over ℚ. Conjecture 2.1 implies that for every rigid prime p, there is an abelian extension K/ℚ of degree p such that

E(ℚ)/pE(ℚ) ≅ Vis_{J}(� (A/ℚ)[p]),

where J = Res_{K/ℚ}(E_{K}) and A ⊂ J has dimension p − 1 and rank 0.

Proof. Conjecture 2.1 produces a prime ℓ ∤ N_{E} with ℓ ≡ 1 (mod p) such that L(E, χ_{p,ℓ}, 1) ≠ 0 and a_{ℓ}(E) ≢ ℓ + 1 (mod p). We will apply Theorem 4.1 to A, B = E, and J = Res_{K/ℚ}(E_{K}) where K is the abelian extension corresponding to χ_{p,ℓ} and A = ker(J ⟶^{tr} E). Since L(E, χ_{p,ℓ}, 1) ≠ 0 and A is attached to the modular form f ⊗ χ_{p,ℓ}, Kato’s work implies that A(ℚ) is finite, so (J/E)(ℚ) is finite. Lemma 3.1 implies that

p ∤ #(J/E)(ℚ) · #E(ℚ)_{tor} · #Φ_{A,ℓ}(𝔽_{ℓ}) · #Φ_{E,ℓ}(𝔽_{ℓ}).

The only hypothesis of Theorem 4.1 left to check is that p ∤ #Φ_{A,q}(𝔽_{q}) for all q | N_{E}. This is because

Φ_{A,q}[p] ⊂ Φ_{A,q}(\overline{𝔽}_{p})[p] ≅ Φ_{A_{K}, q}(\overline{𝔽}_{p})[p] = \{0\},

since K/ℚ is unramified at q (since q ≠ ℓ), formation of Néron models commutes with tame base extensions, A_{K} ≈ ∏ E_{K}, and p is rigid for E. Thus Theorem 4.1 implies that E(ℚ)/pE(ℚ) ≅ Vis_{J}(� (A)[p]).

5 A Connection with the BSD Conjecture

Let E be an elliptic curve and suppose that L(E, 1) = 0. Suppose that (1) we can prove that there is a rigid prime p for E and a prime ℓ as in Conjecture 2.1. Then (2) the rank 0 BSD formula would predict that

� (A/ℚ)[p] ≠ {0}

since^{2} L(A, 1)/Ω_{A} ≡ L(E, 1)/Ω_{E} = 0 (mod ℘) and p ∤ ∏_{q} #Φ_{A,q}(𝔽_{q}). If (3) we also have � (E/K)[p] = {0}, then Vis_{J}(� (A/ℚ)[p]) ≠ {0}, so E(ℚ) is infinite. Example 5.1. Let E be the rank 2 elliptic curve 389A. The prime p = 3 is rigid, and ℓ = 7 satisfies 2.1. We have

(ℤ/3ℤ)^{2} ≅ E(ℚ)/3E(ℚ) ≅ Vis_{J}(� (A/ℚ)[3])

for A ⊂ Res_{K/ℚ}(E_{K}) a rank 0 abelian variety of dimension 2. It would be interesting to investigate the relationship between Kato’s Euler system for A and this subgroup of � (A/ℚ).

6 A Nonsquare Shafarevich-Tate Group

Let E be the rank 1 elliptic curve of conductor 43. Then p = 3 is rigid and ℓ = 7 satisfies Conjecture 2.1. We have

ℤ/3ℤ ≅ E(ℚ)/3E(ℚ) ≅ Vis_{J}(� (A/ℚ)[3])

for the two-dimensional kernel of J → E. Amazingly, the Birch and Swinnerton-Dyer conjecture predicts that #� (A/ℚ)[3] = 3, which is not a perfect square. This is the first abelian variety ever found in which #� (A/ℚ)[p]_{an} is not a perfect square for an odd prime p.

2I actually haven’t written down a proof of this congruence in general, but it should be easy.