Visibility of Mordell-Weil Groups

William A. Stein^{1}

Abstract.

We introduce a notion of visibility for Mordell-Weil groups, make a conjecture about visibility, and support it with theoretical evidence and data. These results shed new light on relations between Mordell-Weil and Shafarevich-Tate groups.

1 Introduction

Consider an exact sequence 0 → C → B → A → 0 of abelian varieties over a number field K. We say that the covering B → A is optimal since its kernel C is connected. As introduced in [LT58], there is a corresponding long exact sequence of Galois cohomology

0 → C(K) → B(K) → A(K) ⟶^{δ} H^{1}(K, C) → H^{1}(K, B) → H^{1}(K, A) → ⋯

The study of the Mordell-Weil group A(K) is central in arithmetic geometry. For example, the Birch and Swinnerton-Dyer conjecture (BSD conjecture) of [Bir71, Tat66]), which is one of the Clay Math Problems [Wil00], asserts that the rank r of A(K) equals the ordering vanishing of L(A, s) at s = 1, and also gives a conjectural formula for L^{(r)}(A, 1) in terms of the invariants of A.

The group H^{1}(K, A) is also of interest in connection with the BSD conjecture, because it contains the Shafarevich-Tate group

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

which is the most mysterious object appearing in the BSD conjecture.

1This material is based upon work supported by the National Science Foundation under Grant No. 0400386.

2 William A. Stein^{2}

Definition 1.0.1 (Visibility). The visible subgroup of H^{1}(K, C) relative to the embedding C ↪ B is

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

The visible quotient of A(K) relative to the optimal covering B → A is

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

We say an abelian variety over ℚ is modular if it is a quotient of the modular Jacobian J_{1}(N) = Jac(X_{1}(N)), for some N. For example, every elliptic curve over ℚ is modular [BCDT01].

This paper gives evidence toward the following conjecture that Mordell-Weil groups should give rise to many visible Shafarevich-Tate groups.

Conjecture 1.0.2. Let A be an abelian variety over a number field K. For every integer m, there is an exact sequence 0 → C → B → A → 0 such that:

1. The image of B(K) in A(K) is contained in mA(K), so A(K)/mA(K) is a quotient of Vis^{B}(A(K)).

2. If K = ℚ and A is modular, then B is modular.

3. The rank of C is zero.

4. We have Coker(B(K) → A(K)) ⊂ � (C/K), via the connecting homomorphism.

In [Ste04] we give the following computational evidence for this conjecture.

Theorem 1.0.3. Let E be the rank 1 elliptic curve y^{2} +y = x^{3} −x of conductor 37. Then Conjecture 1.0.2 is true for all primes m = p < 25000 with p ≠ 2, 37.

Let f = ∑ a_{n}q^{n} be the newform associated to the elliptic curve E of Theorem 1.0.3. Suppose p is one of the primes in the theorem. Then there is an ℓ ≡ 1 (mod p) and a surjective Dirichlet character χ : (ℤ/ℓℤ)^{∗} → μ_{p} such that L(f ⊗ χ, 1) ≠ 0. The C of the theorem is, up to isogeny, the abelian variety associated to f^{χ}, which has dimension p − 1.

In general, we expect the construction of [Ste04] to work for any elliptic curve and any odd prime p of good reduction. The main obstruction to proving that it does work is proving a nonvanishing result for the special values L(f^{χ}, 1). In [Ste04], we verified this hypothesis using modular symbols for p < 25000.

A surprising observation that comes out of the construction of [Ste04] is that #� (A) = p·n^{2}, where n^{2} is an integer square. We thus obtained the first ever examples of abelian varieties whose Shafarevich-Tate groups have order neither a square nor twice a square.

1.1 Contents

In Section 2, we give a brief review of results about visibility of Shafarevich-Tate groups. In Section 3, we give evidence for Conjecture 1.0.2 using results of Kato, Lichtenbaum and Mazur. Section 4 is about bounding the dimension of the abelian varieties in which Mordell-Weil groups are visible. We prove that every Mordell-Weil group is 2-visible relative to an abelian surface. In Section 5, we describe how to construct visible quotients of Mordell-Weil groups, and carry out a computational study of relations between Mordell-Weil groups of elliptic curves and the arithmetic of rank 0 factors of J_{0}(N).

1.2 Acknowledgement

The author had extremely helpful conversations with Barry Mazur and Grigor Grigorov. Proposition 3.0.3 was proved jointly with Ken Ribet. The author was supported by NSF grant DMS-0400386. He used Magma [BCP97] and Python [Ros] for computing the data in Section 5.

2 Review of Visibility of Galois Cohomology

In this section, we briefly review visibility of elements of H^{1}(K, A), as first introduced by Mazur in [CM00, Maz99], and later developed by Agashe and Stein in [Aga99a, AS05, AS02]. We describe two basic results about visibility, and in Section 2.2 we discuss modularity of elements of H^{1}(K, A).

Consider an exact sequence of abelian varieties

0 → A → B → C → 0

over a number field K. Elements of H^{0}(K, C) are points, so they are relatively easy to “visualize”, but elements of H^{1}(K, A) are mysterious.

There is a geometric way to view elements of H^{1}(K, A). The Weil-Chatalet group WC(A/K) of A over K is the group of isomorphism classes of principal homogeneous spaces for A, where a principal homogeneous space is a variety X and a simply-transitive action A×X → X. Thus X is a twist of A as a variety, but X(K) = ∅, unless X is isomorphic to A. Also, the elements of � (A) correspond to the classes of X that have a K_{v}-rational point for all places v. By [LT58, Prop. 4], there is an isomorphism between H^{1}(K, A) and WC(A/K).

In [CM00], Mazur introduced the visible subgroup of H^{1} as in Definition 1.0.1 in order to help unify diverse constructions of principal homogeneous spaces. Many papers were subsequently written about visibility, including [Aga99b, Maz99, Kle01, AS02, MO03, DWS03, AS05, Dum01].

Remark 2.0.1. Note that Vis_{B} H^{1}(K, A) depends on the embedding of A into B. For example, if B = B_{1} × A. Then there could be nonzero visible elements if A is embedded into the first factor, but there will be no nonzero visible elements if A is embedded into the second factor.

A connection between visibility and WC(A/K) is as follows. Suppose

0 → A → B ⟶^{π} C → 0

is an exact sequence of abelian varieties and that c ∈ H^{1}(K, A) is visible in B. Thus there exists x ∈ C(K) such that δ(x) = c, where δ : C(K) → H^{1}(K, A) is the connecting homomorphism. Then X = π^{−1}(x) ⊂ B is a translate of A in B, so the group law on B gives X the structure of principal homogeneous space for A, and this homogeneous space in WC(A/K) corresponds to c.

2.1 Basic Facts

Two basic facts about visibility are that the visible subgroup of H^{1}(K, A) in B is finite, and that each element of H^{1}(K, A) is visible in some B.

Lemma 2.1.1. The group Vis_{B} H^{1}(K, A) is finite.

Proof. Let C = B/A. By the Mordell-Weil theorem C(K) is finitely generated. The group Vis_{B} H^{1}(K, A) is a homomorphic image of C(K) so it is finitely generated. On the other hand, it is a subgroup of H^{1}(K, A), so it is a torsion group. But a finitely generated torsion abelian group is finite.

Proposition 2.1.2. Let c ∈ H^{1}(K, A). Then there exists an abelian variety B and an embedding A ↪ B such that c is visible in B. Moreover, B can be chosen to be a twist of a power of A.

Proof. See [AS02, Prop. 1.3] for a cohomological proof or [JS05, §5] for an equivalent geometric proof. Johan de Jong also proved that everything is visible somewhere in the special case dim(A) = 1 using Azumaya algebras, Néron models, and étale cohomology, as explained in [CM00, pg. 17–18], but his proof gives no (obvious) specific information about the structure of B.

2.2 Modularity

Usually one focuses on visibility of elements in � (A) ⊂ H^{1}(K, A). The papers [CM00, AS02, AS05] contain a number of results about visibility in various special cases, and tables involving elliptic curves and modular abelian varieties.

For example, if A ⊂ J_{0}(389) is the 20-dimensional simple newform abelian variety, then we show that

ℤ/5ℤ × ℤ/5ℤ ≅ E(ℚ)/5E(ℚ) ⊂ � (A),

where E is the elliptic curve of conductor 389. The divisibility 5^{2} | #� (A) is as predicted by the BSD conjecture. The paper [AS05] contains a few dozen other examples like this; in most cases, explicit computational construction of the Shafarevich-Tate group seems hopeless using any other known techniques.

The author has conjectured that if A is a modular abelian variety, then every element of � (A) is modular, i.e., visible in a modular abelian variety. It is a theorem that if c ∈ � (A) has order either 2 or 3 and A is an elliptic curve, then c is modular (see [JS05]).

3 Results Toward Conjecture 1.0.2

The main result of this section is a proof of parts 1 and 2 of Conjecture 1.0.2 for elliptic curves over ℚ. We prove more generally that Mazur’s conjecture on finite generatedness of Mordell-Weil groups over cyclotomic ℤ_{p}-extensions implies part 1 of Conjecture 1.0.2. Then we observe that for elliptic curves over ℚ, Mazur’s conjecture is known, and prove that the abelian varieties that appear in our visibility construction are modular, so parts 1 and 2 of Conjecture 1.0.2 are true for elliptic curves over ℚ.

For a prime p, the cyclotomic ℤ_{p}-extension of ℚ is an extension ℚ_{p^{∞}} of ℚ with Galois group ℤ_{p}; also ℚ_{p^{∞}} is contained in the cyclotomic field ℚ(μ_{p^{∞}}). We let ℚ_{p^{n}} denote the unique subfield of ℚ_{p^{∞}} of degree p^{n} over ℚ. If K is an arbitrary number field, the cyclotomic ℤ_{p}-extension of K is K_{p^{∞}} = K·ℚ_{p^{∞}}. We denote by K_{p^{n}} the unique subfield of K_{p^{∞}} of degree p^{n} over K. The extension K_{p^{∞}} of K decomposes as a tower

K = K_{p^{0}} ⊂ K_{p^{1}} ⊂ ⋯ ⊂ K_{p^{n}} ⊂ ⋯ ⊂ K_{p^{∞}} = ⋃_{n=0}^{∞} K_{p^{n}}.

Mazur hints at the following conjecture in [Maz78] and [RM05, §3]:

Conjecture 3.0.1 (Mazur). If A is an abelian variety over a number field K and p is a prime, then A(K_{p^{∞}}) is a finitely generated abelian group.

Let L/K be a finite extension of number fields and A an abelian variety over K. In much of the rest of this paper we will use the restriction of scalars R = Res_{L/K}(A_{L}) of A viewed as an abelian variety over L. Thus R is an abelian variety over K of dimension [L : K], and R represents the following functor on the category of K-schemes:

S ↦ E_{L}(S_{L}).

If L/K is Galois, then we have an isomorphism of Gal(\overline{ℚ}/K)-modules

R(\overline{ℚ}) = A(\overline{ℚ}) ⊗_{ℤ} ℤ[Gal(L/K)],

where τ ∈ Gal(\overline{ℚ}/K) acts on ∑ P_{σ} ⊗ σ by

τ (∑ P_{σ} ⊗ σ) = ∑ τ(P_{σ}) ⊗ τ_{|L} · σ,

where τ_{|L} is the image of τ in Gal(L/K).

Theorem 3.0.2. Conjecture 3.0.1 implies part 1 of Conjecture 1.0.2. More precisely, if A/K is an abelian variety, m is a positive integer, and A(K_{p^{∞}}) is finitely generated for each p | m, then there is an optimal covering of the form B = Res_{L/K}(A_{L}) → A such that L is abelian over K and the image of B(K) in A(K) is contained in mA(K).

Proof. Fix a prime p | m. Let M = K_{p^{∞}}. Because A(M) is finitely generated, some finite set of generators must be in a single sufficiently large A(K_{p^{n}}), and for this n we have A(M) = A(K_{p^{n}}). For any integer j > 0 let

R_{j} = Res_{K_{p^{j}} /K}(A_{K_{p^{j}}}).

Then, as explained in [Ste04], the trace map induces an exact sequence

0 → B_{j} → R_{j} ⟶^{π_{j}} A → 0,

with B_{j} an abelian variety. Then for any j ≥ n, A(K_{p^{j}}) = A(K_{p^{n}}), so

Vis^{B_{j}}(A(K)) ≅ A(K)/π_{j}(R_{j}(K))
= A(K)/ Tr_{K_{p^{j}} /K}(A(K_{p^{j}}))
= A(K)/ Tr_{K_{p^{n}} /K}(Tr_{K_{p^{j}} /K_{p^{n}}} (A(K_{p^{j}})))
= A(K)/ Tr_{K_{p^{n}} /K}(Tr_{K_{p^{j}} /K_{p^{n}}} (A(K_{p^{n}})))
= A(K)/ Tr_{K_{p^{n}} /K}(p^{j−n}A(K_{p^{n}}))
= A(K)/p^{j−n} Tr_{K_{p^{n}} /K}(A(K_{p^{n}}))
→ A(K)/p^{j−n}A(K),

where the last map is surjective since

Tr_{K_{p^{n}} /K}(A(K_{p^{n}})) ⊂ A(K).

Arguing as above, for each prime p | m, we find an extension L_{p} of K of degree a power of p such that Tr_{L_{p}/K}(A(L_{p})) ⊂ p^{ν_{p}}A(K), where ν_{p} = ord_{p}(m). Let L be the compositum of the fields L_{p}. Then for each p | m,

Tr_{L/K}(A(L)) = Tr_{L_{p}/K}(Tr_{L/L_{p}}(A(L))) ⊂ Tr_{L_{p}/K}(A(L_{p})) ⊂ p^{ν_{p}}A(K).

Thus

Tr_{L/K}(A(L)) ⊂ ⋂_{p|m} p^{ν_{p}}A(K) = mA(K),  (1)

where for the last equality we view A(K) as a finite direct sum of cyclic groups.

Let R = Res_{L/K}(A_{L}). Then trace induces an optimal cover R → A, and (1) implies that we have the required surjective map

Vis^{R}(A(K)) = A(K)/ Tr_{L/K}(A(L)) → → A(K)/mA(K).

We will next prove parts 1 and 2 of Conjecture 1.0.2 for elliptic curves over ℚ by observing that Conjecture 3.0.1 is a theorem of Kato in this case. We first prove a modularity property for restriction of scalars. Recall that a modular abelian variety is a quotient of J_{1}(N).

Proposition 3.0.3. If A is a modular abelian variety over ℚ and K is an abelian extension of ℚ, then Res_{K/ℚ}(A_{K}) is also a modular abelian variety.

Proof. Since A is modular, A is isogenous to a product of abelian varieties A_{f} attached to newforms in S_{2}(Γ_{1}(N)), for various N. Since the formation of restriction of scalars commutes with products, it suffices to prove the proposition under the hypothesis that A = A_{f} for some newform f. Let R = Res_{K/ℚ}(A_{f}). As discussed in [Mil72, pg. 178], for any prime p there is an isomorphism of ℚ_{p}-adic Tate modules

V_{p}(R) ≅ Ind_{G_{K}}^{G_{ℚ}} V_{p}(A_{K}).

The induced representation on the right is the direct sum of twists of V_{p}(A_{K}) by characters of Gal(K/ℚ). This is isomorphic to the ℚ_{p}-adic Tate module of some abelian variety P = ∏_{χ} A_{g^{χ}}, where χ runs through certain Dirichlet characters corresponding to the abelian extension K/ℚ, and g runs through certain Gal(\overline{ℚ}/ℚ)-conjugates of f, and g^{χ} denotes the twist of g by χ. Falting’s theorem (see e.g., [Fal86, §5]) then gives us the desired isogeny R → P.

It is not necessary to use the full power of Falting’s theorem to prove this proposition, since Ribet [Rib80] gave a more elementary proof of Falting’s theorem in the case of modular abelian varieties. However, we must work some to apply Ribet’s theorem, since we do not know yet that R is modular.

Let R and P be as above. Over \overline{ℚ}, the abelian variety A is isogenous to a power of a simple abelian variety B, since if more than one non-isogenous simple occurred in the decomposition of A/\overline{ℚ}, then End(A/\overline{ℚ}) would not be a matrix ring over a (possibly skew) field (see [Rib92, §5]). For any character χ, by the (3) ⟹ (2) assertion of [Rib80, Thm. 4.7], the abelian varieties A_{f} and A_{f^{χ}} are isogenous over \overline{ℚ} to powers of the same abelian variety A′, hence to powers of the simple B. A basic property of restriction of scalars is that R_{K} is isomorphic to a power of (A_{f})_{K}, hence R_{K} is isogenous over \overline{ℚ} to a power of B. Thus R and P are both isogenous over \overline{ℚ} to a power of B, so R is isogenous to P over \overline{ℚ}, since they have the same dimension, as their Tate modules are isomorphic. Let L be a Galois number field over which such an isogeny is defined. Consider the natural Gal(\overline{ℚ}/ℚ)-equivariant inclusion

Hom(R_{ℚ}, P_{ℚ})⊗_{ℚ_{p}} ↪ Hom_{Gal(\overline{ℚ}/ℚ)}(V_{p}(R), V_{p}(P)).  (2)

By Ribet’s proof of the Tate conjecture for modular abelian varieties [Rib80], the inclusion

Hom(R_{L}, P_{L})⊗_{ℚ_{p}} ↪ Hom_{Gal(\overline{ℚ}/L)}(V_{p}(R), V_{p}(P))  (3)

is an isomorphism, since there is an isogeny P_{L} → R_{L} and P is modular. But then (2) must also be an isomorphism, since (2) is the result of taking Gal(L/ℚ)-invariants of both sides of (3).

By construction of P, there is an isomorphism V_{p}(R) ≅ V_{p}(P) of Gal(\overline{ℚ}/ℚ)- modules, so by (2) there is an isomorphism in Hom(R_{ℚ}, P_{ℚ})⊗ℚ_{p}. Thus there is a ℚ_{p}-linear combination of elements of Hom(R_{ℚ}, P_{ℚ}) that has nonzero determinant. However, if a ℚ_{p}-linear combination of matrices has nonzero determinant, then some ℚ-linear combination does, since the determinant is a polynomial function of the coefficients and ℚ is dense in ℚ_{p}. Thus there is an isogeny R → P defined over ℚ, so R is modular.

Corollary 3.0.4. Parts 1 and 2 of Conjecture 1.0.2 are true for every elliptic curve E over ℚ.

Proof. Suppose p is a prime, and let ℚ_{p^{∞}} be the cyclotomic ℤ_{p} extension of ℚ. By [BCDT01], E is a modular elliptic curve, so Rohrlich [Roh84] implies that all but finitely many special values L(E, χ, 1) are nonzero, where χ runs over all Dirichlet characters of p-power order. Kato proved (see, e.g., [Kat04, Sch98]) that if L(E, χ, 1) ≠ 0, then the χ part of E(ℚ_{p^{∞}})⊗ℚ vanishes. Combining these results, we see that E(ℚ_{p^{∞}}) is finitely generated, so we can apply Theorem 3.0.2 to conclude that if x ∈ E(ℚ) and m | order(x), then x is m-visible relative to an optimal cover of E by a restriction of scalars B from an abelian extension. Then Proposition 3.0.3 implies that B is modular.

4 The Visibility Dimension

The visibility dimension is analogous to the visibility dimension for elements of H^{1}(K, A) introduced in [AS02, §2]. We prove below that elements of order 2 in Mordell-Weil groups of elliptic curves over ℚ are 2-visible relative to an abelian surface. Along the way, we make a general conjecture about stability of rank and show that it implies a general bound on the visibility dimension.

Definition 4.0.5 (Visibility Dimension). Let A be an abelian variety over a number field K and suppose m is an integer. Then A has m-visibility dimension n if there is an optimal cover B → A with n = dim(B) and the image of B(K) in A(K) is contained in mA(K), so A(K)/mA(K) is a quotient of Vis^{B}(A(K)).

The following rank-stability conjecture is motivated by its usefulness for proving a result about m-visibility.

Conjecture 4.0.6. Suppose A is an abelian variety over a number field K, that L is a finite extension of K, and m > 0 is an integer. Then there is an extension M of K of degree m such that rank(A(K)) = rank(A(M)) and M ∩ L = K.

The following proposition describes how Conjecture 4.0.6 can be used to find an extension where the index of A(K) in A(M) is coprime to m.

Proposition 4.0.7. Let A be an abelian variety over a number field K and suppose m is a positive integer. If Conjecture 4.0.6 is true for A and m, then there is an extension M of K of degree m such that A(M)/A(K) is of order coprime to m.

Proof. Choose a finite set P_{1}, …, P_{n} of generators for A(K). Let

L = K (\frac{1}{m}P_{1}, …, \frac{1}{m}P_{n})

be the extension of K generated by all mth roots of each P_{i}. Since the set of mth roots of a point is closed under the action of Gal(\overline{K}/K), the extension L/K is Galois. Note also that the m torsion of A is defined over L, since the differences of conjugates of a given \frac{1}{m}P_{i} are exactly the elements of A[m]. Let S be the set of primes of K that ramify in L.

By our hypothesis that Conjecture 4.0.6 is true for A and m, there is an extension M of K of degree m such that

rank(A(K)) = rank(A(M))

and M ∩ L = K. In particular, C = A(M)/A(K) is a finite group. Suppose, for the sake of contradiction, that gcd(m, #C) ≠ 1, so there is some prime divisor p | m and an element [Q] ∈ C of exact order p. Here Q ∈ A(M) is such that pQ ∈ A(K) but Q ∉ A(K). Because P_{1}, …, P_{n} generate A(K) and pQ ∈ A(K), there are integers a_{1}, … a_{n} such that

pQ = ∑_{i=1}^{n} a_{i}P_{i}.

Then for any fixed choice of the \frac{1}{p}P_{i}, we have

Q − ∑_{i=1}^{n} a_{i} · \frac{1}{p}P_{i} ∈ A[p],

since

p (Q − ∑_{i=1}^{n} a_{i} · \frac{1}{p}P_{i}) = pQ − ∑_{i=1}^{n} a_{i} · P_{i} = 0.

Thus Q ∈ A(L). But then since L ∩ M = K, so we obtain a contradiction from

Q ∈ A(L) ∩ A(M) = A(K).

With Proposition 4.0.7 in hand, we show that Conjecture 4.0.6 bounds the visibility dimension of Mordell-Weil groups. In particular, we see that Conjecture 4.0.6 implies that for any abelian variety A over a number field K, and any m, there is an embedding A(K)/mA(K) ↪ H^{1}(K, C) coming from a δ map, where C is an abelian variety over K of rank 0.

Theorem 4.0.8. Let A be an abelian variety over a number field K and suppose m is a positive integer. If Conjecture 4.0.6 is true for A and m, then there is an optimal covering B → A with B of dimension m such that

Vis^{B}(A(K)) ≅ A(K)/mA(K).

Proof. By Proposition 4.0.7, there is an extension M of K of degree m such that the quotient A(M)/A(K) is finite of order coprime to m. Then, as in [Ste04], the restriction of scalars B = Res_{M/K}(A_{M}) is an optimal cover of A and

Vis^{B}(A(K)) ≅ A(K)/ Tr(A(M)).

However, there is also an inclusion A ↪ B from which one sees that

mA(M) ⊂ Tr(A(M)),

so Vis^{B}(A(K)) is an m-torsion group.

We have

[Tr(A(M)) : Tr(A(K))] | [A(M) : A(K)].

We showed above that gcd([A(M) : A(K)], m) = 1, so since

Tr(A(M))/ Tr(A(K))

is killed by m, it follows that Tr(A(M)) = Tr(A(K)). We conclude that

Vis^{B}(A(K)) = A(K)/mA(K).

Proposition 4.0.9. If E is an elliptic curve over ℚ and m = 2, then Conjecture 4.0.6 is true for E and m.

Proof. Let L be as in Conjecture 4.0.6, so L is an extension of ℚ of possibly large degree. Let D be the discriminant of L. By [MM97, BFH90] there are infinitely many quadratic imaginary extensions M of ℚ such that L(E^{M}, 1) ≠ 0, where E^{M} is the quadratic twist of E by M. By [Kol91, Kol88] all these curves have rank 0. Since there are only finitely many quadratic fields ramified only at the primes that divide D, there must be some field M that is ramified at a prime p ∤ D. If M is contained in L, then all the primes that ramify in M divide D, so M is not contained in L. Since M is quadratic, it follows that M ∩ L = ℚ, as required. Since the image of E(ℚ) + E^{M}(ℚ) in E(M) has finite index, it follows that E(M)/E(ℚ) is finite.

Corollary 4.0.10. If E is an elliptic curve over ℚ, then there is an optimal cover B → E, with B a 2-dimension modular abelian variety, such that

Vis^{B}(E(ℚ)) ≅ E(ℚ)/2E(ℚ).

Proof. Combine Proposition 4.0.9 with Theorem 4.0.8. Also B is modular since it is isogenous to E × E′, where E′ is a quadratic twist of E.

Note that the B of Corollary 4.0.10 is isomorphic to (E × E^{D})/Φ, where E^{D} is a rank 0 quadratic imaginary twist of E and Φ ≅ E[2] is embedded antidiagonally in E × E^{D}. Note that E^{D} also has analytic rank 0, since it was constructed using the theorems of [Kol91, Kol88] and [MM97, BFH90]. Thus our construction is compatible with the one of Proposition 5.1.1 below.

5 Some Data About Visibility and Modularity

This section contains a computational investigation of modularity of Mordell-Weil groups of elliptic curves relative to abelian varieties that are quotients of J_{0}(N). One reason that we restrict to J_{0}(N) is so that computations are more tractable. Also, for m > 2, the twisting constructions that we have given in previous sections are no longer allowed since they take place in J_{1}(N). Furthermore, the work of [KL89] suggests that we understand the arithmetic of J_{0}(N) better than that of J_{1}(N).

5.1 A Visibility Construction for Mordell-Weil Groups

The following proposition is an analogue of [AS02, Thm. 3.1] but for visibility of Mordell-Weil groups (compare also [CM00, pg. 19]).

Proposition 5.1.1. Let E be an elliptic curve over a number field K, and let Φ = E[m] as a Gal(\overline{K}/K)-module. Suppose A is an abelian variety over K such that Φ ⊂ A, as G_{ℚ}-modules. Let B = (A × E)/Φ, where Φ is embedded anti-diagonally. Then there is an exact sequence

0 → B(K)/(A(K) + E(K)) → E(K)/mE(K) → Vis^{B}(E(K)) → 0.

Moreover, if (A/E[m])(K) is finite of order coprime to m, then the first term of the sequence is 0, so

Vis^{B}(E(K)) ≅ E(K)/mE(K).

Proof. Using the definition of B and multiplication by m on E, we obtain the following commutative diagram, whose rows and columns are exact:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & E[m] \ar[r] \ar[d] & E \ar[r]^{m} \ar[d] & E \ar[d]^{≅} \ar[r] & 0 \\ 0 \ar[r] & A \ar[r] \ar[d] & B \ar[r] \ar[d] & E \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A/E[m] \ar[r]^{≅} \ar[d] & B/E \ar[r] \ar[d] & 0 \\  & 0 & 0}

Taking K-rational points we arrive at the following diagram with exact rows and columns:

\xymatrix{0 \ar[r] & E(K)/E(K)[m] \ar[r]^{m} \ar[d] & E(K) \ar[r] \ar[d]^{≅} & E(K)/mE(K) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & B(K)/A(K) \ar[r] \ar[d] & E(K) \ar[r] \ar[d] & Vis^{B}(E(K)) \ar[r] & 0 \\  & B(K)/(A(K) + E(K)) & 0}

The snake lemma and the fact that the middle vertical map is an isomorphism implies that the right vertical map is a surjection with kernel isomorphic to B(K)/(A(K) + E(K)). Thus we obtain an exact sequence

0 → B(K)/(A(K) + E(K)) → E(K)/mE(K) → Vis^{B}(E(K)) → 0.

This proves the first statement of the proposition. For the second, note that we have an exact sequence 0 → E → B → A/E[m] → 0. Taking Galois cohomology yields an exact sequence

0 → E(K) → B(K) → (A/E[m])(K) → ⋯,

so #(B(K)/E(K)) | #(A/E[m])(K). If (A/E[m])(K) is finite of order coprime to m, then B(K)/(A(K) + E(K)) has order dividing #(A/E[m])(K), so the quotient B(K)/(A(K) + E(K)) is trivial, since it injects into E(K)/mE(K).

5.2 Tables

The data in this section suggests the following conjecture.

Conjecture 5.2.1. Suppose E is an elliptic curve over ℚ and p is a prime such that E[p] is irreducible. Then there exists infinitely many newforms g ∈ S_{2}(Γ_{0}(N)), for various integers N, such that L(g, 1) ≠ 0 and E[p] ⊂ A_{g} and Vis^{B}(E(ℚ)) = E(ℚ)/pE(ℚ), where B = (A_{g} × E)/E[p].

Let E be the elliptic curve y^{2} + y = x^{3} − x. This curve has conductor 37 and Mordell-Weil group free of rank 1. According to [Cre97], E is isolated in its isogeny class, so each E[p] is irreducible.

Table 1 gives for each N the odd primes p such that there is a mod p congruence between f_{E} and some newform g in S_{2}(Γ_{0}(37N)) such that A_{g} has rank 0 and the isogeny class of A_{g} contains no abelian variety with rational p torsion. The first time a p occurs, it is in bold. We bound the torsion in the isogeny class using the algorithm from [AS05, §3.5] with primes up to 17. Thus by Proposition 5.1.1, the Mordell-Weil group of E is p-modular of level 37N. A − means there are no such p. Table 2, which was derived directly from Table 1, gives for a prime p, all integers N such that E(ℚ) is p-modular of level 37N.

Table 1: Visibility of Mordell-Weil for y^{2} + y = x^{3} − x

\begin{matrix}N & p′s \\ 2 & 5 \\ 3 & 7 \\ 4 & − \\ 5 & − \\ 6 & − \\ 7 & 3 \\ 8 & − \\ 9 & − \\ 10 & − \\ 11 & 17 \\ 12 & − \\ 13 & − \\ 14 & − \\ 15 & − \\ 16 & − \\ 17 & 3 \\ 18 & − \end{matrix}  \begin{matrix}N & p′s \\ 19 & 5 \\ 20 & − \\ 21 & 7 \\ 22 & − \\ 23 & 11 \\ 24 & − \\ 25 & − \\ 26 & − \\ 27 & 3 \\ 28 & − \\ 29 & 3 \\ 30 & − \\ 31 & 3 \\ 32 & − \\ 33 & 7 \\ 34 & − \\ 35 & − \end{matrix}  \begin{matrix}N & p′s \\ 36 & − \\ 37 & − \\ 38 & 5 \\ 39 & − \\ 40 & − \\ 41 & 3, 17 \\ 42 & − \\ 43 & 7 \\ 44 & − \\ 45 & − \\ 46 & − \\ 47 & 3 \\ 48 & − \\ 49 & − \\ 50 & 5 \\ 51 & − \\ 52 & − \end{matrix}  \begin{matrix}N & p′s \\ 53 & 53 \\ 54 & − \\ 55 & − \\ 56 & − \\ 57 & − \\ 58 & − \\ 59 & 13 \\ 60 & − \\ 61 & 5, 7 \\ 62 & − \\ 63 & 3 \\ 64 & − \\ 65 & − \\ 66 & − \\ 67 & 3, 5 \\ 68 & − \\ 69 & − \end{matrix}  \begin{matrix}N & p′s \\ 70 & − \\ 71 & 3, 7 \\ 72 & − \\ 73 & 3, 5 \\ 74 & − \\ 75 & − \\ 76 & − \\ 77 & − \\ 78 & − \\ 79 & − \\ 80 & − \\ 81 & 3 \\ 82 & − \\ 83 & 3, 11 \\ 84 & − \\ 85 & − \\ 86 & − \end{matrix}  \begin{matrix}N & p′s \\ 87 & − \\ 88 & − \\ 89 & 43 \\ 90 & − \\ 91 & 3 \\ 92 & − \\ 93 & 7 \\ 94 & − \\ 95 & − \\ 96 & − \\ 97 & 47 \\ 98 & − \\ 99 & − \\ 100 & − \\ 101 & 3, 11 \\ 102 & − \\ 103 & 43 \end{matrix}  \begin{matrix}N & p′s \\ 104 & − \\ 105 & − \\ 106 & 5 \\ 107 & 3, 5 \\ 108 & − \\ 109 & 3, 7 \\ 110 & − \\ 111 & − \\ 112 & − \\ 113 & 3, 11 \\ 114 & − \\ 115 & − \\ 116 & − \\ 117 & − \\ 118 & − \\ 119 & 3 \\ 120 & − \end{matrix}

\begin{matrix}N & p′s \\ 121 & − \\ 122 & − \\ 123 & − \\ 124 & − \\ 125 & 5 \\ 126 & − \\ 127 & 127 \\ 128 & − \\ 129 & − \\ 130 & − \\ 131 & 3 \\ 132 & − \\ 133 & − \\ 134 & − \\ 135 & − \\ 136 & − \\ 137 & 3 \end{matrix}  \begin{matrix}N & p′s \\ 138 & − \\ 139 & 17 \\ 140 & − \\ 141 & − \\ 142 & − \\ 143 & − \\ 144 & − \\ 145 & − \\ 146 & − \\ 147 & 7 \\ 148 & − \\ 149 & 5, 31 \\ 150 & − \\ 151 & 17 \\ 152 & − \\ 153 & 3 \\ 154 & − \end{matrix}  \begin{matrix}N & p′s \\ 155 & − \\ 156 & − \\ 157 & 3, 5 \\ 158 & − \\ 159 & − \\ 160 & − \\ 161 & − \\ 162 & − \\ 163 & 7, 13 \\ 164 & − \\ 165 & − \\ 166 & − \\ 167 & 3, 5 \\ 168 & − \\ 169 & − \\ 170 & − \\ 171 & − \end{matrix}  \begin{matrix}N & p′s \\ 172 & − \\ 173 & 3, 5, 11 \\ 174 & − \\ 175 & − \\ 176 & − \\ 177 & − \\ 178 & − \\ 179 & 3 \\ 180 & − \\ 181 & 3, 59 \\ 182 & − \\ 183 & − \\ 184 & − \\ 185 & − \\ 186 & − \\ 187 & − \\ 188 & − \end{matrix}  \begin{matrix}N & p′s \\ 189 & 3 \\ 190 & − \\ 191 & 7 \\ 192 & − \\ 193 & 5, 11 \\ 194 & − \\ 195 & − \\ 196 & − \\ 197 & 3, 5, 13 \\ 198 & − \\ 199 & 3, 11 \\ 200 & − \\ 201 & − \\ 202 & 5 \\ 203 & 3 \\ 204 & − \\ 205 & − \end{matrix}  \begin{matrix}N & p′s \\ 206 & − \\ 207 & − \\ 208 & − \\ 209 & − \end{matrix}

14 William A. Stein^{8}

Table 2: Levels Where Mordell-Weil is p-Visible for y^{2} + y = x^{3} − x p N such that 37N is a level of p-modularity of E(ℚ)

3  7, 17, 27, 29, 31, 41, 47, 63, 67, 71, 73, 81, 83, 91, 101, 107, 109, 113, 119, 131, 137, 153, 157, 167, 173, 179, 181, 189, 197, 199, 203
5  {2, 19, 38, 50, 61, 67, 73, 106, 107, 125, 149, 157, 167, 173, \atop 193, 197, 202}
7  3, 21, 33, 43, 61, 71, 93, 109, 147, 163, 191
11  23, 83, 101, 113, 173, 193, 199
13  59, 163, 197
17  11, 41, 139, 151
19 − 29 -
31 149
37 − 41 -
43  89, 103
47  97
53  53
59  181
61 − 113 -
127  127

Table 3: Visibility of Mordell-Weil for y^{2} + y = x^{3} + x^{2}

\begin{matrix}N & p′s \\ 2 & 5 \\ 3 & 3 \\ 4 & − \\ 5 & 5 \\ 6 & − \\ 7 & − \\ 8 & − \\ 9 & − \\ 10 & − \\ 11 & 3 \\ 12 & − \\ 13 & 19 \\ 14 & − \\ 15 & − \\ 16 & − \end{matrix}  \begin{matrix}N & p′s \\ 17 & 3, 7 \\ 18 & − \\ 19 & − \\ 20 & − \\ 21 & − \\ 22 & 5 \\ 23 & 5 \\ 24 & − \\ 25 & − \\ 26 & − \\ 27 & 3 \\ 28 & − \\ 29 & 3 \\ 30 & − \\ 31 & − \end{matrix}  \begin{matrix}N & p′s \\ 32 & − \\ 33 & 3 \\ 34 & 5 \\ 35 & − \\ 36 & − \\ 37 & 19 \\ 38 & − \\ 39 & 3 \\ 40 & − \\ 41 & 37 \\ 42 & − \\ 43 & − \\ 44 & − \\ 45 & − \\ 46 & − \end{matrix}  \begin{matrix}N & p′s \\ 47 & − \\ 48 & − \\ 49 & − \\ 50 & 5 \\ 51 & 3 \\ 52 & − \\ 53 & 59 \\ 54 & − \\ 55 & 5 \\ 56 & − \\ 57 & 3 \\ 58 & − \\ 59 & 3 \\ 60 & − \\ 61 & 5 \end{matrix}  \begin{matrix}N & p′s \\ 62 & − \\ 63 & − \\ 64 & − \\ 65 & − \\ 66 & − \\ 67 & 71 \\ 68 & − \\ 69 & − \\ 70 & − \\ 71 & 5, 7 \\ 72 & − \\ 73 & 3 \\ 74 & − \\ 75 & − \\ 76 & − \end{matrix}  \begin{matrix}N & p′s \\ 77 & − \\ 78 & − \\ 79 & − \\ 80 & − \\ 81 & 3 \\ 82 & − \\ 83 & 3, 23 \\ 84 & − \\ 85 & 5 \\ 86 & − \\ 87 & 3 \\ 88 & − \\ 89 & 47 \\ 90 & − \\ 91 & − \end{matrix}  \begin{matrix}N & p′s \\ 92 & − \\ 93 & − \\ 94 & − \\ 95 & − \\ 96 & − \\ 97 & 7, 13 \\ 98 & − \\ 99 & 3 \\ 100 & − \end{matrix}

Ribet’s level raising theorem [Rib90] gives necessary and sufficient conditions on a prime N for there to be a newform g of level 37N that is congruent to f_{E} modulo p. Note that the form g is new rather than just p-new since 37 is prime and there are no modular forms of level 1 and weight 2. If, moreover, we impose the condition L(g, 1) ≠ 0, then Ribet’s condition requires that p divides N + 1 + εa_{N}, where ε is the root number of E. Since E has odd analytic rank, in this case ε = −1. For each primes p ≤ 127 and each N ≤ 203, were find the levels of such g. The only cases in which we don’t already find a congruence level already listed in Table 2 corresponding to a newform with torsion multiple coprime to p are

p = 3, N = 43  and  p = 19, N = 47, 79.

In all other cases in which Ribet’s theorem produces a congruent g with ord L(g, s) even (hence possibly 0), we actually find a g with L(g, 1) ≠ 0 and can show that #A_{g}(ℚ)_{tor} is coprime to p.

For p = 3 and N = 43 we find a unique newform g ∈ S_{2}(Γ_{0}(1591)) that is congruent to f_{E} modulo 3. This form is attached to the elliptic curve y^{2} + y = x^{3} − 71x + 552 of conductor 1591, which has Mordell-Weil groups ℤ ⊕ ℤ. Thus this is an example of a congruence relating a rank 1 curve to a rank 2 curve. For p = 19 and N = 47, the g has degree 43, so A_{g} has dimension 43, we have L(g, 1) ≠ 0, but the torsion multiple is 76 = 19 · 4, which is divisible by 19. For p = 19 and N = 79, the A_{g} has dimension 57, we have L(g, 1) ≠ 0, but the torsion multiple is 76 again.

Tables 3–4 are the analogues of Tables 1–2 but for the elliptic curve y^{2}+y = 16 William A. Stein^{9}

Table 4: Levels Where Mordell-Weil is p-Visible for y^{2} + y = x^{3} + x^{2} p N such that 43N is a level of p-modularity of E(ℚ)

3 3, 11, 17, 27, 29, 33, 39, 51, 57, 59, 73, 81, 83, 87, 99
5 2, 5, 22, 23, 34, 50, 55, 61, 71, 85
7 17, 71, 97
11 -
13 97
17 -
19 13, 37
23 83
29, 31 -
37 41
41, 43 -
47 89
53 -
59 53
61, 67 -
71 67

Table 5: Visibility of Mordell-Weil for y^{2} + y = x^{3} + x^{2} − 2x

\begin{matrix}N & p′s \\ 1 & 5 \\ 2 & − \\ 3 & − \\ 4 & − \\ 5 & 3 \\ 6 & − \end{matrix}  \begin{matrix}N & p′s \\ 7 & 3 \\ 8 & − \\ 9 & 3 \\ 10 & − \\ 11 & − \\ 12 & − \end{matrix}  \begin{matrix}N & p′s \\ 13 & 11 \\ 14 & − \\ 15 & 3 \\ 16 & − \\ 17 & − \\ 18 & − \end{matrix}  \begin{matrix}N & p′s \\ 19 & − \\ 20 & − \\ 21 & − \\ 22 & − \\ 23 & 5 \\ 24 & − \end{matrix}  \begin{matrix}N & p′s \\ 25 & − \\ 26 & − \\ 27 & 3 \\ 28 & − \\ 29 & 3 \end{matrix}

Table 6: Levels Where Mordell-Weil is p-Visible for y^{2} + y = x^{3} + x^{2} − 2x p N such that 389N is a level of p-modularity of E(ℚ)

3 5, 7, 9, 15, 27, 29
5 1, 23
7 -
11 13

x^{3} + x^{2} of conductor 43. This elliptic curve also has rank 1 and all mod p representations are irreducible. The primes p and N such that Ribet’s theorem produces a congruent g with ord_{s=1} L(g, s) even, yet we do not find one with L(g, 1) ≠ 0 and the torsion multiple coprime to p are

p = 3, N = 31, 61  and  p = 11, N = 19, 31, 47, 79.

The situation for p = 11 is interesting since in this case all the g with ord_{s=1} L(g, s) even fail to satisfy our hypothesis. At level 19 · 43 we find that g has degree 18 and L(g, 1) ≠ 0, but the torsion multiple is divisible by 11.

Let E be the elliptic curve y^{2} + y = x^{3} + x^{2} − 2x of conductor 389. This curve has Mordell-Weil group free of rank 2. Tables 5–6 are the analogues of Tables 1–2 but for E. The primes p and N such that Ribet’s theorem produces a congruent g with ord_{s=1} L(g, s) even, yet we do not find one with L(g, 1) ≠ 0 and the torsion multiple coprime to p are

p = 3, N = 17  and  p = 5, N = 19.

For p = 3, there is a unique g of level 6613 = 37 · 17 with ord_{s=1} L(g, s) even and E[3] ⊂ A_{g}. This form has degree 5 and L(g, 1) = 0, so this is another example where the rank 0 hypothesis of Proposition 5.1.1 is not satisfied. Note that the torsion multiple in this case is 1. For p = 5, there is a unique g of level 7391 = 37 · 19, with ord_{s=1} L(g, s) even and E[5] ⊂ A_{g}. This form has degree 4 and L(g, 1) ≠ 0, but the torsion multiple is divisible by 5.

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William A. Stein

Department of Mathematics Harvard University Cambridge, MA 02138

           was@math.harvard.edu