Visibility of Shafarevich-Tate Groups at Higher Level

William Stein

               http://modular.fas.harvard.edu

April 19, 2004

Abstract

I will begin by introducing the Birch and Swinnerton-Dyer conjecture in the context of abelian varieties attached to modular forms, and discuss some of the main results about it. I will then introduce Mazur’s notion of visibility of Shafarevich-Tate groups and explain some of the basic facts and theorems. Cremona, Mazur, Agashe, and myself carried out large computations about visibility for modular abelian varieties of level N in J_{0}(N). These computations addressed the following question: If A is a modular abelian variety of level N, how much of the Shafarevich-Tate group � (A) is modular of level N, i.e., visible in J_{0}(N). The results of these computations suggest that often much of the Shafarevich-Tate group is not modular of level N. It is then natural to ask if every element of � (A) is modular of level M, for some multiple M = NR, and if so, what can one say about the set of such M? I will finish the talk with some new data and a conjecture about this last question, which is still very much open.

1 Modular Abelian Varieties

Let N be a positive integer and consider the congruence subgroup

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

(Almost everything in this talk also makes sense with Γ_{0}(N) replaced by Γ_{1}(N).) The modular curve

X_{0}(N) = Γ_{0}(N) \ ({z ∈ C : Im(z) > 0} ∪ Q ∪ {∞})

is a Riemann surface that is the set of complex points of an algebraic curve over Q. We will not use that

X_{0}(N)(C) = { isomorphism classes of (E, C) } ∪ { cusps } .

Our primary interest is the Jacobian

J_{0}(N) = Jac(X_{0}(N))

which is an abelian variety over Q of dimension equal to the genus of X_{0}(N). The points on the Jacobian parametrize, in a natural way, the divisor classes of degree 0 on X_{0}(N).

Let S_{2}(Γ_{0}(N)) be the cusp forms of weight 2 for Γ_{0}(N). This is the finite-dimensional complex vector space of holomorphic functions on the upper half plane such that

f(z)dz = f(γ(z))d(γ(z))

for all γ ∈ Γ_{0}(N), and which “vanish at the cusps”. The map f(z) ↦ f(z)dz induces

S_{2}(Γ_{0}(N)) ≅ H^{0}(X_{0}(N)_{C}, Ω^{1})

so S_{2}(Γ_{0}(N)) has dimension the genus of X_{0}(N).

The Hecke algebra is a commutative ring

T = Z[T_{1}, T_{2}, T_{3}, …]

which acts on S_{2}(Γ_{0}(N)) and J_{0}(N). A newform

f = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{2}(Γ_{0}(N))

is an eigenvector for every element of T normalized so a_{1} = 1, which does not “come from” any lower level. Attached to f there is an ideal

I_{f} = Ann_{T}(f) = Ker(T → Z[a_{1}, a_{2}, …]),

and (following Shimura) to this ideal we attach an abelian variety A_{f} and an L-function L(A_{f}, s).

Let

A_{f} = J_{0}(N)[I_{f}]^{0} = ( ⋂_{φ∈I_{f}} Ker(φ))^{0}

be the connected component of the intersections of the kernels of elements of I_{f}. Then A_{f} has dimension [K_{f} : Q] = [Q(a_{1}, a_{2}, …) : Q)], and is define over Q.

Let

L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s)

where d = [K_{f} : Q] and the f_{i} are the Galois conjugates of f. Also,

L(f, s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} .

Hecke proved that L(f, s) is entire and satisfies a functional equation.

The abelian varieties A_{f} are a rich class of abelian varieties. The elliptic curves over Q are all isogenous to some A_{f} (the Wiles-Breuil-Conrad-Diamond-Taylor modularity theorem).

2 The Birch and Swinnerton-Dyer Conjecture

2.1 Conjecture

Conjecture 2.1 (Birch and Swinnerton-Dyer).

1. rank A_{f}(Q) = ord_{s=1} L(A_{f}, s)

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

Remarks: Part of the conjecture is that � (A_{f}) is finite. There is also a conjecture for arbitrary abelian varieties over global fields. Clay Math Problem: $1000000 prize for proof of (1) in case dim(A_{f}) = 1

Here:

• c_{p} is the Tamagawa number at the prime p, and the product is over the prime divisors of N.

• Ω_{A_{f}} is the canonical Néron measure of A_{f}(R).

• Reg_{A_{f}} is the regulator (absolute value of Néron-Tate canonical height pairing matrix).

• A_{f}(Q)_{tor} is the torsion subgroup of A_{f}(Q).

• � (A_{f}) is the Shafarevich-Tate group.

2.2 Evidence

• Rubin: results in CM Case

• Kolyvagin, Logachev, Gross-Zagier, et al.: If ord_{s=1} L(f, s) = 0 or 1, then (1) true and � (A_{f}) finite.

• Cremona: Compute � (A_{f})_{?} (=conjectural order) for tens of thousands of A_{f} of dimension 1 and get approximate square order. (Theorem of Cassels: if E an elliptic curve and � (E) finite then order a perfect square. Note that the analogue for abelian varieties is false; for exampe, I’ve constructed examples for each odd prime p < 25000 of abelian varieties A of dimension p − 1 such that � (A) = p · n^{2}.)

In this talk I will focus on A_{f} of possibly large dimension with L(A_{f}, 1) ≠ 0, since computation of Reg_{A_{f}} is difficult (impossible?) when one can’t even reasonably hope to write down A_{f} explicitly with equations.

3 Visibility of Shafarevich-Tate Groups

3.1 Definitions

It is easy to write down a point on an elliptic curve E. You simply write down a pair of rational numbers, which are a solution to a Weierstrass equation. In contrast, imagine describing explicitly an element of � (E) of order 2003. The most direct way would be to give a genus one curve (with principal homogeneous space structure), embedded in P^{3} of degree at least 2003 (!), hence very complicated.

The idea of visibility of Shafarevich-Tate groups was introduced by Barry Mazur around 1998 to unify various constructions of elements of Shafarevich-Tate groups.

Definition 3.1 (Shafarevich-Tate Group).

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

Here H^{1}(K, A) is the first Galois cohomology, which can be interpreted geometrically as the Weil-Chatalet group

WC(A/K) = { principal homogenous spaces X for A }/ ∼ .

Then � (A) is the subgroup of locally trivial classes of homogenous spaces. For example

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

Fix an inclusion i : A ↪ B of abelian varieties and let π : B → C be the quotient of B by the image of A, so we have an exact sequence

0 → A → B → C → 0

of abelian varieties.

Definition 3.2 (Visible Subgroup).

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

and

Vis_{i}(� (A)) = Ker(� (A) → � (B)).

1. The visible subgroup is finite because B(K) is finitely generated and Vis_{i}(H^{1}(K, A)) is torsion.

2. If c ∈ Vis_{i}(H^{1}(K, A)), then c is also “visible” in the sense that if c is the image of a point x ∈ C(K), and if X = π^{−1}(x) ⊂ B, then [X] ∈ WC(A) corresponds to c.

3. The visibile subgroups depends on the choice of embedding i : A ↪ B. I’ve also considered defining Vis_{B}(H^{1}(K, A)) to be the subgroup generated by all visible subgroups with respect to all embeddings A → B, but I’m not sure what properties this definition has.

3.2 Theorems

“Everything is visible somewhere.”

Theorem 3.3 (Stein). If c ∈ H^{1}(K, A) then there exists B = Res_{L/K}(A_{L}) such that i : A ↪ B and c ∈ Vis_{i}(H^{1}(K, A)). (Here L is such that res_{L/K}(c) = 0.)

“Visibility construction.”

Theorem 3.4 (Agashe-Stein). Suppose A, B ⊂ C over Q, that A + B = C, that A ∩ B is finite. Suppose N is divisible by all bad primes for C, and p is a prime such that

• B[p] ⊂ A

• p ∤ 2 · N · #B(Q)_{tor} · #(C/B)(Q)_{tor} · ∏_{p|N} c_{A,p} · c_{B,p}.

If A has rank 0, then there is a natural inclusion

B(Q)/pB(Q) ↪ Vis_{C}(� (A)).

(And certain generalizations...)

3.3 Example

Example 3.5. For N = 389, take B the (first ever) rank 2 elliptic curve, and A the 20dimensional rank 0 factor.

\xymatrix{ & B \ar[d] \\ A \ar[r] & J_{0}(389)}

Gives

(Z/5Z)^{2} ≅ B(Q)/5B(Q) ↪ � (A).

Part 2 of the Birch and Swinnerton-Dyer conjecture predicts that

� (A) = 5^{2} · 2^{?},

so this gives evidence.

4 Visibility in Modular Jacobians

Suppose now A = A_{f} ⊂ J_{0}(N) is attached to a newform.

Definition 4.1 (Modular of level M). An element c ∈ � (A)[p] is modular of level M if c ∈ Vis_{M}^{p}(� (A)), where Vis_{M}^{p}(� (A)) is the subgroup generated by all kernels of maps � (A)[p^{∞}] → � (J_{0}(M))[p^{∞}] induced by homomorphisms A → J_{0}(M) of degree coprime to p.

Note that M must be a multiple of N.

Question 4.2 (Mazur). Suppose E ⊂ J_{0}(N) is an elliptic curve of conductor N. How much of � (E) is modular of level N?

Answer: In examples, surprisingly much. Expect not all visible, since

Vis_{N}(� (E)) ⊂ � (E)[modular degree],

and modular degree annihilates symmetric square Selmer group (work of Flach).

4.1 Data and Experiments

• Cremona-Mazur: There are 52 elliptic curves E ⊂ J_{0}(N) with N < 5500 such that p | #� (E)_{?}. Cremona-Mazur show that for 43 of these that � (E) “probably” is modular of level N, and for 3 that it is definitely not: N = 2849, 4343, 5389. (“Probably” was made “provably” in many cases in subsequent work.)

• Agashe-Stein: Same question as Cremona-Mazur for A_{f} ⊂ J_{0}(N) of any dimension. Using results of my Ph.D. thesis, MAGMA packages, etc. I computed a divisor and multiple of #� (A_{f})_{?} for the following:

– 10360 abelian varieties A_{f} ⊂ J_{0}(N) with L(A_{f}, 1) ≠ 0.

– Found 168 with #� (A_{f})_{?} definitely divisible by an odd prime.

– For 39 of these, prove that all #� (A_{f})_{?}^{odd} elements are modular of level N, and 106 probably are. This gives strong evidence for the BSD conjecture, and a sense that maybe something further is going on.

– Of these 168, at least 62 have odd conjectural � that is definitely not modular of level N. Big mystery? Where is this � modular? Is it modular at all? Is it even there?? (Perhaps a good place to look for counterexample to BSD.)

5 Visibility at Higher Level

Definition 5.1. Let c ∈ � (A_{f}). The modularity levels of c are the set of integers

N (c) = {M : c ∈ Vis_{M}(� (A_{f}))}.

Conjecture 5.2 (Stein). For any c ∈ � (A_{f}) we have

N (c) ≠ ∅,

i.e., every element of � (A_{f}) is modular.

Motivation: This is a working hypothesis that makes computing with modular abelian varieties easier. Also, if there were a common level at which all of � (A_{f}) were modular, then � (A_{f}) would be finite, and conversely (assuming the conjecture).

5.1 Ribet Level Raising

Suppose that f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) is a newform and 𝔭 is a nonzero prime ideal of Z[a_{1}, a_{2}, …] such that A_{f}[𝔭] is irreducible. If

a_{ℓ} + ℓ + 1 ≡ 0 (mod 𝔭)

then there exists an ℓ-newform g ∈ S_{2}(Γ_{0}(Nℓ)) such that i(A_{f}[𝔭]) = A_{g}[𝔭] for an appropriate i : J_{0}(N) → J_{0}(Nℓ) of degree coprime to char(𝔭) and the sign of the functional equations for L(f, s) and L(g, s) are the same.

If we instead require that a_{ℓ} −(ℓ+1) ≡ 0 (mod 𝔭) then there is such a g, but the sign of the functional equation changes, and the new Tamagawa numbers of A_{g} at ℓ will (or tends to be?) divisible by 𝔭.

5.2 Evidence for Conjecture

I defined a precise notion of “probably modular” motivated by Theorem 3.4 and what I can compute. In many cases I could do extra work and actually prove modularity; however, at this stage it is more interesting to gather data to see what is going on, in order to have a sense for what to conjecture.

Mazur proved that everything in � (E)[3], for E an elliptic curve, is visible in an abelian surface, which, together with the modularity theorem, might imply modularity of � (E)[3] at higher level. Same for 2, proved by me and by a different method by Thomas Klenke.

6 Some Tables

The first two pages of tables below give some of the data that I computed about visibility of Shafarevich-Tate groups at level N. The third table gives the new data about visibility at higher level.

Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim S_{l}  S_{u} moddeg(A)^{odd} B dim A^{∨} ∩ B̃^{∨} Vis
389E∗ 20 5^{2}  =  5  389A 1  [20^{2}]  5^{2}
433D∗ 16 7^{2}  =  7·111  433A 1  [14^{2}]  7^{2}
446F∗ 8 11^{2} =  11·359353  446B 1  [11^{2}] 11^{2}
551H 18 3^{2}  =  169  NONE
563E∗ 31 13^{2} =  13  563A 1  [26^{2}] 13^{2}
571D∗ 2 3^{2}  =  3^{2}·127  571B 1  [3^{2}]  3^{2}
655D∗ 13 3^{4}  =  3^{2}·9799079  655A 1  [36^{2}]  3^{4}
681B 1 3^{2}  =  3·125  681C 1  [3^{2}]  −
707G∗ 15 13^{2} =  13·800077  707A 1  [13^{2}] 13^{2}
709C∗ 30 11^{2} =  11  709A 1  [22^{2}] 11^{2}
718F∗ 7 7^{2}  =  7·5371523  718B 1  [7^{2}]  7^{2}
767F 23 3^{2}  =  1  NONE
794G∗ 12 11^{2} = 11·34986189  794A 1  [11^{2}]  −
817E∗ 15 7^{2}  =  7·79  817A 1  [7^{2}]  −
959D 24 3^{2}  =  583673  NONE
997H∗ 42 3^{4}  =  3^{2}  997B 1  [12^{2}]  3^{2}
997C 1  [24^{2}]  3^{2}
1001F 3 3^{2}  =  3^{2}·1269  1001C 1  [3^{2}]  −
91A 1  [3^{2}]  −
1001L 7 7^{2}  =  7·2029789  1001C 1  [7^{2}]  −
1041E 4 5^{2}  =  5^{2}·13589  1041B 2  [5^{2}]  −
1041J 13 5^{4}  = 5^{3}·21120929983 1041B 2  [5^{4}]  −
1058D 1 5^{2}  =  5·483  1058C 1  [5^{2}]  −
1061D 46 151^{2} =  151·10919  1061B 2 [2^{2}302^{2}] −
1070M 7 3·5^{2} 3^{2}·5^{2} 3·5·1720261 1070A 1  [15^{2}]  −
1077J 15 3^{4}  = 3^{2}·1227767047943 1077A 1  [9^{2}]  −
1091C 62 7^{2}  =  1  NONE
1094F∗ 13 11^{2} = 11^{2}·172446773 1094A 1  [11^{2}] 11^{2}
1102K 4 3^{2} = 3^{2}·31009 1102A 1 [3^{2}] −
1126F∗ 11 11^{2} = 11·13990352759 1126A 1  [11^{2}] 11^{2}
1137C 14 3^{4}  = 3^{2}·64082807  1137A 1  [9^{2}]  −
1141I 22 7^{2}  =  7·528921  1141A 1  [14^{2}]  −
1147H 23 5^{2}  =  5·729  1147A 1  [10^{2}]  −
1171D∗ 53 11^{2} =  11·81  1171A 1  [44^{2}] 11^{2}
1246B 1 5^{2}  =  5·81  1246C 1  [5^{2}]  −
1247D 32 3^{2}  =  3^{2}·2399  43A 1  [36^{2}]  −
1283C 62 5^{2}  =  5·2419  NONE
1337E 33 3^{2}  =  71  NONE
1339G 30 3^{2}  =  5776049  NONE
1355E 28 3  3^{2} 3^{2}·2224523985405 NONE
1363F 25 31^{2} =  31·34889  1363B 2 [2^{2}62^{2}] −
1429B 64 5^{2}  =  1  NONE
1443G 5 7^{2}  =  7^{2}·18525  1443C 1 [7^{1}14^{1}] −
1446N 7 3^{2}  =  3·17459029  1446A 1  [12^{2}]  −

Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim S_{l} S_{u} moddeg(A)^{odd}  B dim A^{∨} ∩ B̃^{∨} Vis
1466H∗ 23 13^{2} = 13·25631993723 1466B 1  [26^{2}]  13^{2}
1477C∗ 24 13^{2} = 13·57037637  1477A 1  [13^{2}]  13^{2}
1481C 71 13^{2} =  70825  NONE
1483D∗ 67 3^{2}·5^{2} =  3·5  1483A 1  [60^{2}] 3^{2}·5^{2}
1513F 31  3  3^{4}  3·759709  NONE
1529D 36  5^{2}  =  535641763  NONE
1531D 73  3  3^{2}  3  1531A 1  [48^{2}]  −
1534J 6  3  3^{2} 3^{2}·635931  1534B 1  [6^{2}]  −
1551G 13  3^{2}  = 3·110659885  141A 1  [15^{2}]  −
1559B 90 11^{2} =  1  NONE
1567D 69 7^{2}·41^{2} =  7·41  1567B 3 [4^{4}1148^{2}] −
1570J∗ 6  11^{2} = 11·228651397  1570B 1  [11^{2}]  11^{2}
1577E 36  3  3^{2}  3^{2}·15  83A 1  [6^{2}]  −
1589D 35  3^{2}  = 6005292627343  NONE
1591F∗ 35 31^{2} =  31·2401  1591A 1  [31^{2}]  31^{2}
1594J 17  3^{2}  = 3·259338050025131 1594A 1  [12^{2}]  −
1613D∗ 75  5^{2}  =  5·19  1613A 1  [20^{2}]  5^{2}
1615J 13  3^{4}  = 3^{2}·13317421  1615A 1 [9^{1}18^{1}]  −
1621C∗ 70 17^{2} =  17  1621A 1  [34^{2}]  17^{2}
1627C∗ 73  3^{4}  =  3^{2}  1627A 1  [36^{2}]  3^{4}
1631C 37  5^{2}  = 6354841131  NONE
1633D 27 3^{6}·7^{2} = 3^{5}·7·31375  1633A 3 [6^{4}42^{2}]  −
1634K 12  3^{2}  = 3·3311565989  817A 1  [3^{2}]  −
1639G∗ 34 17^{2} =  17·82355  1639B 1  [34^{2}]  17^{2}
1641J∗ 24 23^{2} = 23·1491344147471 1641B 1  [23^{2}]  23^{2}
1642D∗ 14  7^{2}  = 7·123398360851 1642A 1  [7^{2}]  7^{2}
1662K 7  11^{2} = 11·16610917393 1662A 1  [11^{2}]  −
1664K 1  5^{2}  =  5·7  1664N 1  [5^{2}]  −
1679C 45 11^{2} =  6489  NONE
1689E 28  3^{2}  = 3·172707180029157365 563A 1  [3^{2}]  −
1693C 72 1301^{2} =  1301  1693A 3 [2^{4}2602^{2}] −
1717H∗ 34 13^{2} =  13·345  1717B 1  [26^{2}]  13^{2}
1727E 39  3^{2}  =  118242943  NONE
1739F 43 659^{2} = 659·151291281 1739C 2 [2^{2}1318^{2}] −
1745K 33  5^{2}  = 5·1971380677489 1745D 1  [20^{2}]  −
1751C 45  5^{2}  =  5·707  103A 2 [505^{2}]  −
1781D 44  3^{2}  =  61541  NONE
1793G∗ 36 23^{2} = 23·8846589  1793B 1  [23^{2}]  23^{2}
1799D 44  5^{2}  =  201449  NONE
1811D 98 31^{2} =  1  NONE
1829E 44 13^{2} =  3595  NONE
1843F 40  3^{2}  =  8389  NONE
1847B 98  3^{6}  =  1  NONE
1871C 98 19^{2} =  14699  NONE

Visibility at Higher Level

A_{f} with odd invisible � _{an}[ℓ] All ℓ-congruent

A_{g} ⊂ J_{0}(Np)_{new} with Np ≤ 5000 and ord_{s=1} L(g, s) ≥ 0 (and higher Np if known)

551, dim 18, ℓ = 3  p = 2: dim 1, rank 2

p = 3: dim 1, rank 2 p = 5: dim 25, rank 0

767, dim 23, ℓ = 3  p = 2: dim 1, rank 2

p = 7: dim 1, rank 2 p = 7: dim 52, rank 0

959, dim 24, ℓ = 3  p = 2: dim 1, rank 2
1091, dim 62, ℓ = 7  p = 7: dim 2, rank 2
1283, dim 62, ℓ = 5  p = 3: dim 2, rank 2
1337, dim 33, ℓ = 3  p = 2: dim 1, rank 2
1339, dim 30, ℓ = 3  p = 2: dim 1, rank 2
1355, dim 28, ℓ = 3  p = 2: dim 1, rank 2
1429, dim 64, ℓ = 5  p = 2: dim 2, rank 2

p = 3: dim 66, rank 0

1481, dim 71, ℓ = 13  Nothing in range
1513, dim 31, ℓ = 3  p = 2: dim 1, rank 2
1529, dim 36, ℓ = 5  p = 7: dim 1, rank 2
1559, dim 90, ℓ = 11  Nothing in range
1589, dim 35, ℓ = 3  Nothing in range
1631, dim 37, ℓ = 5  p = 2: dim 1, rank 2
1679, dim 45, ℓ = 11  p = 2: dim 2, rank 2
1727, dim 39, ℓ = 3  p = 2: dim 1, rank 2
2849, dim 1, ℓ = 3  p = 3: dim 1, rank 2
4343, dim 1, ℓ = 3  Nothing in range
5389, dim 1, ℓ = 3  p = 7: dim 1, rank 2

When the second column contains an A_{g} of rank 2, then � (A_{f})[ℓ] is “very likely” to be visible of level M = Np. This is the case for most examples. The “Nothing in range” note means that the smallest p for which there exists g of even analytic rank congruent to f is beyond the range of my current tables. The examples of level 2849, 4343, and 5389 are the odd and definitely invisible examples in Cremona and Mazur’s original paper on visibility.