Nonvanishing Twists and Visible Shafarevich-Tate Groups^{∗}

William A. Stein October 1, 2001

Let E be an elliptic curve over ℚ. Our long-term goal is to find, for many primes p, rank 0 abelian varieties A such that

0 → A → J → E → 0

induces an isomorphism

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

Such results are useful in connecting the rank 0 BSD formula to the conjecture that ord_{s=1} L(E, s) = rk(E).

1 Terminology

This should be enough to help you guess my notation:

• Φ_{E,p} denotes the component group of the Néron model of E at p.

• N_{E} is the conductor of E.

• A prime p is rigid for E if

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

and

ρ_{E,p} : G_{ℚ} → Aut(E[p])

is irreducible.

∗These are notes for a Modular Curves Seminar talk.

2 Visibility Theory

Visibility theory has been developed by Barry Mazur, Amod Agashe, and myself, with periodic help from Brian Conrad.

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

Vis_{J}(� (A)) = ker(� (A) → � (J)).

Theorem 2.1. Suppose A, B ⊂ J, and (A ∩ B)(\overline{ℚ}) is finite. 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]).

Remark 2.2. Everything divides infinity, so if (J/B)(ℚ) is infinite, then no primes satisfy the hypothesis of the theorem.

For the proof, look at [Agashe-Stein, Visibility of Shafarevich-Tate Groups of Abelian Varieties]. It uses the snake lemma, and a careful local analysis at each prime that uses standard arithmetic geometry tools.

3 A Conjecture About Nonvanishing of Twists

Fix E and suppose p is rigid for E. For every ℓ ≡ 1 (mod p), fix

χ_{p,ℓ} : (ℤ/ℓℤ)^{∗} → → μ_{p}

of order p and conductor ℓ.

Conjecture 3.1 (–). There exists a prime ℓ ∤ N_{E} such that

L(E, χ_{p,ℓ}, 1) ≠ 0

and

a_{ℓ}(E) ≢ ℓ + 1 (mod p).

Evidence: The conjecture is true for every pair (E, p) I’ve tried, e.g., for all rigid p < 50 for the first 20 rank 1 optimal quotients of J_{0}(N) and the first two rank 2 quotients.

The following “Density Conjecture” will not be needed for our application:

Conjecture 3.2 (–). The set of primes ℓ ≡ 1 (mod p) such that L(E, χ_{p,ℓ}, 1) = 0 has Dirichlet density 0 amongst all primes.

4 p-Torsion

Fix

• elliptic curve E

• rigid prime p

• a prime ℓ ≡ 1 (mod p) such that ℓ ∤ N_{E}.

Let K/ℚ be the abelian extension corresponding to a character χ_{p,ℓ} : (ℤ/ℓℤ)^{∗} → μ_{p} of order p and conductor ℓ.

The diagram we will plug into visibility theory is:

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

Michael Stoll helped me to prove the following lemma.

Lemma 4.1. If a_{ℓ}(E) ≢ 2 (mod p), then the following groups have no nontrivial p-torsion:

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

Proof.

• We first that prove

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. The action of Frob_{ℓ} ∈ Gal(ℚ_{ℓ}^{ur}/ℚ_{ℓ}) on

E[p](ℚ_{ℓ}^{ur}) = E[p](\overline{ℚ}_{ℓ})

has characteristic polynomial

x^{2} − a_{ℓ}(E)x + ℓ ∈ 𝔽_{p}[x].

This polynomial does not have +1 as a root, so

E[p](ℚ_{ℓ}) = 0.

If z ∈ E[p](K_{v}) then ℚ_{ℓ}(z) ⊂ K_{v} and K_{v} is totally ramified, so ℚ_{ℓ}(z) = ℚ_{ℓ} and z = 0. Thus J(ℚ_{ℓ})[p] = 0.

• Next,

(J/E)(ℚ_{ℓ}) ⊂ (J/E)(K_{v}) ≈ E(K_{v}) × ⋯ × E(K_{v}),

so (J/E)(ℚ_{ℓ})[p] = 0.

• Finally, consider Φ_{A,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.

5 Visualizing Mordell-Weil in Rank 0 Sha

Theorem 5.1. Let E be an elliptic curve over ℚ. Conjecture 3.1 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 3.1 produces a prime ℓ ≡ 1 (mod p) such that L(E, χ_{p,ℓ}, 1) ≠ 0 and a_{ℓ}(E) ≢ 2 (mod p). Since L(E, χ_{p,ℓ}, 1) ≠ 0 and A is attached to f ⊗ χ_{p,ℓ}, Kato’s work implies that A(ℚ) is finite. Lemma 4.1 implies that

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

To apply Theorem 2.1, we just need that p ∤ #Φ_{A,p}(𝔽_{p}). This is true becaue Φ_{A,p}(\overline{𝔽}_{p}) = Φ_{A_{K},℘}(\overline{𝔽}_{p}) = 0, since K/ℚ is unramified at p, and A_{K} = E_{K} × ⋯ × E_{K} and E has good reduction at p. Thus

E(ℚ)/pE(ℚ) ≅ Vis_{J}(� (A)[p]),

as claimed.

BSD Connection: Let E be an elliptic curve. Suppose we don’t know anything about E(ℚ), but do know that L(E, 1) = 0. If we could prove that there is a rigid prime such that

� (A/ℚ)[p] ≠ 0 (as better be predicted by the BSD formula)

and

� (E/K)[p] = 0,

then Theorem 5.1 would imply that E(ℚ) is infinite.

6 Example

Let E be the rank 2 curve \underline{389A}. The prime p = 3 is rigid, and ℓ = 7 satisfies 3.1. We have

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

for A ⊂ Res_{K/ℚ}(E_{K}) of rank zero.