Visibility of Shafarevich-Tate groups

William A. Stein^{∗}

Fall 2000

Abstract These are notes for my talks on visibility of Shafarevich-Tate groups of quotients of J_{0}(N). I first introduce the notion of visibility in the context of trying to understand the Birch and Swinnerton-Dyer conjecture. I then give some experimental data, which suggests what one might expect. Next I sketch the theory behind proving that some of Sha is visible. Finally, I discuss visibility for quotients of J_{1}(N), and for motives attached to modular forms.

1 Modular abelian varieties

A central class of abelian varieties of interest in number theory are those which are modular, in these sense that they are a quotient of some modular Jacobian J_{0}(N), or, more generally, of some J_{1}(N). For example, we now know that every elliptic curve over Q is a member of this class (see [?]). 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. Attach to I_{f} the 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 one has the following formula:

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

∗email: was@math.harvard.edu

where the right hand side should be interpreted as 0 if A(Q) is infinite. (When L(A, 1) = 0, there is a refinement involving the regulator of A and the leading coefficient of the power series expansion of L(A, s) about s = 1.)

For a nice summary of this question, and some of its generalization in the special case when A has dimension 1, see Wiles’s short article at the CMI web page [?].

2 Mazur: Visualize Sha!

Let A be an abelian variety over Q. To give an element of � (A) is equivalent to 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? They can be found 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, we reveal 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. Cochains are continuous, so 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 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 A. 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.

With the help of A. Agashe, 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 L(A_{f}, 1) ≠ 0 and the property that there is an odd prime ℓ that divides #� (A_{f}). Of these, the full odd part of � (A_{f}) is visible in 22 cases, whereas it is invisible in each 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

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 do not 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!

The tables

Shafarevich-Tate groups of abelian subvarieties of J_{0}(p)

(The entries in the columns “mod deg” and “� _{an}” are only really the odd parts of “mod deg” and “� _{an}”.)

A  dim � _{an}(A)  mod deg(A)  B  dim mod deg(B)
389E  20  5^{2}  5  389A  1  5
433D 16  7^{2}  3 · 7 · 37  433A  1  7
563E  31  13^{2}  13  563A  1  13
571D  2  3^{2}  3^{2} · 127  571B  1  3
709C 30  11^{2}  11  709A  1  11
997H 42  3^{4}  3^{2}  997B  1  3
1061D 46  151^{2}  61 · 151 · 179  1061B  2  151
1091C 62  7^{2}  1  NONE
1171D 53  11^{2}  3^{4} · 11  1171A  1  11
1283C 62  5^{2}  5 · 41 · 59  NONE
1429B 64  5^{2}  1  NONE
1481C 71  13^{2}  5^{2} · 2833  NONE
1483D 67  3^{2} · 5^{2}  3 · 5  1483A  1  3 · 5
1531D 73  3^{2}  3  1531A  1  3
1559B 90  11^{2}  1  NONE
1567D 69  7^{2} · 41^{2}  7 · 41  1567B  3  7 · 41
1613D 75  5^{2}  5 · 19  1613A  1  5
1621C 70  17^{2}  17  1621A  1  17
1627C 73  3^{4}  3^{2}  1627A  1  3^{2}
1693C 72  1301^{2}  1301  1693A  3  1301
1811D 98  31^{2}  1  NONE
1847B 98  3^{6}  1  NONE
1871C 98  19^{2}  14699  NONE
1877B 86  7^{2}  1  NONE
1907D 90  7^{2}  3 · 5 · 7 · 11  1907A  1  7
1913B 1  3^{2}  3 · 103  1913A  1  3 · 5^{2}
1913E 84  5^{4} · 61^{2}  5^{2} · 61 · 103  1913A,C 1, 2 3 · 5^{2}, 5^{2} · 61
1933C 83  3^{2} · 7^{2}  3 · 7  1933A  1  3 · 7
1997C 93  17^{2}  1  NONE
2027C 94  29^{2}  29  2027A  1  29
2029C 90 5^{2} · 269^{2}  5 · 269  2029A  2  5 · 269
2039F 99  3^{4} · 5^{2}  19 · 29 · 7759 · 3214201 NONE
2063C 106  13^{2}  61 · 139  NONE
2089J 91  11^{2}  3 · 5 · 11 · 19 · 73 · 139 2089B  1  11
2099B 106  3^{2}  1  NONE
2111B 112  211^{2}  1  NONE
2113B 91  7^{2}  1  NONE
2161C 98  23^{2}  1  NONE
2333C 101 83341^{2}  83341  2333A  4  83341
2339C 114  3^{8}  6791  NONE
2411B 123  11^{2}  1  NONE
2593B 109 67^{2} · 2213^{2}  67 · 2213  2593A  4  67 · 2213

email: was@math.harvard.edu

Odd “invisible” � in Cremona-Mazur

E  #� _{E} mod deg(E) F  Remarks
2849A  3^{2}  2^{5} · 5 · 61  NONE visible at level 3 · 2849
3364C  7^{2}  2^{6} · 3^{2} · 5^{2} · 7 none  visible in 3-dim F
4229A  3^{2}  2^{3} · 3 · 7 · 13 none
4343B  3^{2}  2^{4} · 1583  NONE
4914N  3^{2}  2^{4} · 3^{5}  none  E has rational 3-torsion
5054C  3^{2}  2^{3} · 3^{3} · 11  none
5073D  3^{2}  2^{5} · 3 · 5 · 7 · 23 none
5389A  3^{2}  2^{2} · 2333  NONE visible at level 7 · 5389

Modular motives of weight 4: Conjecturally visible �

Suppose f, g ∈ S_{4}(Γ_{1}(N)) are such that L(f, 2)/Ω_{f} = p^{2} · ∗, L(g, 2) = 0, and f and g satisfy a “p-congruence”. Does this force p to divide #� (M_{f} (2))?

χ  M_{f}  dim  p^{2}  M_{g} dim
(1, 1)  99C  8  19^{2}  99B  2
1  127C 17  43^{2}  127A 1
1  159E  8  23^{2}  159B 1
(0, 0, 1) 200E  8  7^{2}  200G 2
1  365E 18  29^{2}  365A 1
1  369I  ?  13^{2}  369A ?
1  453E  ?  17^{2}  453A ?
1  465H  ?  11^{2}  465A ?
1  477L  ?  73^{2}  477A ?
1  567G  ?  13^{2}, 23^{2} 567A ?
1  581E  ?  19^{2}  581A ?

Here χ is the common “nebentypus” character of f and g.

Visibility of Mordell-Weil in J_{0}(N) via an odd congruence

L(f, 1) = 0 dim L(g, 1) ≠ 0 dim cong L(Ag , 1)/ΩAg · 2∗
389A  1  389E  20  5  25/97
433A  1  433D  16  7  49/9
446B  1  446F  8  11  121/3
563A  1  563E  31  13  169/281
571B  1  571D  2  3  9
643A  1  INV
655A  1  655D  13  9  81
664A  1  664F  8  5  25
681C  1  681B  1  3  9
707A  1  707G  15  13  169
718B  1  718F  7  7  49
794A  1  794G  12  11  121/3
817A  1  817E  15  7  49/5
916C  1  916G  9  11  121
944E  1  944O  6  7  49
997B  1  997H  42  3  81/83
997C  1  997H  42  3  81/83
1001C  1  1001F  3  3  27
1001C  1  1001L  7  7  49
1028A  1  1028E  14  11  3267
1034A  1  INV
1041B  2  1041E  4  5  25
1041B  2  1041J  13  25  625
1058C  1  1058D  1  5  25
1061B  2  1061D  46  151  22801/265
1070A  1  1070M  7  15  75
1073A  1  INV
1077A  1  1077J  15  9  81
1088J  1  1088R  2  3  9
1094A  1  1094F  13  11  121/3
1102A  1  1102K  4  3  9
1126A  1  1126F  11  11  121
1132A  1  1132F  12  5  225
1137A  1  1137C  14  9  81
1141A  1  1141I  22  7  1524537/41

1143C  1  1143J  9  11  121
1147A  1  1147H  23  5  225/19
1171A  1  1171D  53  11  121/195
1246C  1  1246B  1  5  25
1309B  1  INV
1324A  1  1324E  14  9  6561
1325E  1  1325T  11  9  2187
1363B  2  1363F  25  31  961/5
1431A  1  1431L  14  3  9
1436A  1  INV
1443C  1  1443G  5  7  49
1446A  1  1446N  7  3  9
1466B  1  1466H  23  13  4331806939187/367
1477A  1  1477C  24  13  169
1480A  1  1480G  5  7  49
1483A  1  1483D  67  15  225/247
1525C  1  1525O  16  7  49
1531A  1  1531D  73  3  3/85
1534B  1  1534J  6  3  3
1567B  3  1567D  69  287  82369/261
1570B  1  1570J  6  11  121
1576A  1  1576E  14  11  121
1591A  1  1591F  35  31  6727/19
1594A  1  1594J  17  3  3370648239/19
1608A  1  1608J  6  13  169
1611D  1  1611O  11  9  81
1613A  1  1613D  75  5  25/403
1615A  1  1615J  13  9  102141
1621A  1  1621C  70  17  289/135
1627A  1  1627C  73  9  81/271
1633A  3  1633D  27  189  35721
1639B  1  1639G  34  17  680017/25
1641B  1  1641J  24  23  529
1642A  1  1642D  14  7  49

643A, 1034A, 1073A, 1309B all have modular degree a power of 2; 1436A has modular degree divisible by 3.