The Birch and Swinnerton-Dyer

formula for modular abelian varieties ^{∗}

Amod Agashe University of California, Berkeley May 22, 2000

∗These transperencies can be obtained from http://www.math.berkeley.edu/˜ amod/mymath.html

The Birch and Swinnerton-Dyer conjectural formula

J_{0}(N) = Jacobian of the modular curve X_{0}(N)
T = Hecke algebra
f = a newform s.t. L(f, 1) ≠ 0
I_{f} = Ann_{T}f, an ideal of T
A = A_{f} = J_{0}(N)/I_{f}J_{0}(N),

the Shimura quotient associated to f

Conjecture 1 (Birch, Swinnerton-Dyer, Tate, Gross).

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

where

Ω(A) = real volume of A w.r.t.

Néron differentials

� (A) = Shafarevich-Tate group of A
c_{p}(A) = order of the arithmetic component

group of A at p

A formula for L(A, 1)/Ω(A)

f_{1}, f_{2}, …, f_{d} = Galois conjugates of f

Φ : H_{1}(X_{0}(N), Q)^{+} → C^{d} is given by
γ ↦ {∫_{γ} f_{1}, …, ∫_{γ} f_{d}}

Theorem 2 (Conjectured by Stein).

\frac{L_{A}(1)}{Ω_{A}} = \frac{[Φ(H_{1}(X_{0}(N), Z)^{+} ) : Φ(Te)]}{c_{A} · c_{∞}(A)} ,

where e ∈ H_{1}(X_{0}(N), Q)^{+} is the winding element [Φ(H_{1}(X_{0}(N), Z)^{+} ) : Φ(Te)] = lattice index

c_{A}  = the (generalized) Manin constant
= index of the subgroup of

Néron differentials in S_{2}(Γ_{0}(N), Z)[I_{f}]

c_{∞}(A) = [A(R) : A^{0}(R)]

Remark 3. The right-hand side of the formula above is a rational number and can be computed up to the constant c_{A} using modular symbols; thus one can extend some of the BSD calculations of Cremona for elliptic curves to higher dimensional quotients.

The Manin constant

Theorem 4 (Mazur, Stein, Abbes, Ullmo, A. A. ). If p is a prime s.t. p|c_{A}, then either p^{2} |N, or p = 2 and 2 divides the order of the kernel of

the composite A^{∨} ↪ J_{0}(N) ↠ A.

I = an ideal of T s.t. T/I is torsion-free
B = J_{0}(N)/IJ_{0}(N)
B = Néron model of B over Z

Definition 5. The (generalized) Manin constant is the order of the cokernel of the map H^{0}(B, Ω_{B/Z}) → S_{2}(Γ_{0}(N), Z)[I].

Conjecture 6 (Stein, A. A. ). If B is an optimal quotient of the new quotient of J_{0}(N), then its generalized Manin constant is 1.

Remark 7. The hypothesis above is necessary: c_{J0(33)} = 3 (Edixhoven).

Visible elements of the Shafarevich-Tate group

C = an abelian subvariety of J_{0}(N)

Definition 8 (Mazur). The visible part of � (C) is the kernel of the map � (C) → � (J_{0}(N)) .

Theorem 9 (Mazur, Stein). Let C be an abelian subvariety of J_{0}(N) of rank 0. Suppose D is another abelian subvariety of J_{0}(N) of rank > 0. Let p be a prime s.t. D[p] ⊆ C. Then, under certain mild conditions, p divides the order of the visible part of � (C).

Let B be a quotient of the new part of J_{0}(N). Consider the composite B^{∨} ↪ J_{0}(N) ↠ B, which is an isogeny; call it ϕ_{B}.

Lemma 10 (Mazur, A. A. ). Every visible element of � (B^{∨}) is killed by multiplication by the exponent of ker ϕ_{B}.

Prime levels with #� _{an}(A_{f}) > 1

(Calculated by William Stein) Warning: only odd parts of the invariants are shown .

A_{f}  #� _{an}(A_{f})  √deg(ϕ_{Af}) B_{g}
389E  5^{2}  5  389A
433D  7^{2}  3 · 7 · 37  433A
563E  13^{2}  13  563A
571D  3^{2}  3^{2} · 127  571B
709C  11^{2}  11  709A
997H  3^{4}  3^{2}  997B
1061D  151^{2}  61 · 151 · 179 1061B
1091C  7^{2}  1  NONE
1171D  11^{2}  3^{4} · 11  1171A
1283C  5^{2}  5 · 41 · 59  NONE

⋯

2111B 211^{2} 1 NONE

⋯

2333C 83341^{2} 83341 2333A

⋯

#� _{an}(A_{f}) = order of the Shafarevich-Tate group

as predicted by the BSD formula.

B_{g} = an optimal quotient of J_{0}(N) of positive analytic rank s.t. if an odd prime p divides #� _{an}(A_{f}), then B_{g}^{∨}[p] ⊆ A_{f}^{∨}.

Example 11 (Stein). 5^{2} |#� _{389E} .

Congruence primes and the modular degree

E = strong modular elliptic curve of level N ϕ_{E} : X_{0}(N) → E is its modular parametrization . Modular degree of E = deg ϕ_{E} Congruence number of E = r_{E} =largest integer r s.t. ∃ a modular form g

with integral Fourier coefficients,

orthogonal to f w.r.t. Petersson inner product, and satisfying g ≡ f mod r.

Theorem 12 (Ribet[Zagier] ). deg ϕ_{E} |r_{E}; moreover, if N is prime, then deg ϕ_{E} = r_{E} . If p is a prime s.t. p| \frac{r_{E}}{deg ϕ_{E}}, then p|N.

Question 13 (Frey, Muller). Is deg ϕ_{E} = r_{E}?

Answer: No. E.g. (Stein), when E = 54B1, deg ϕ_{E} = 2

and 3|r_{E}.

Question 14. If p is a prime such that p^{2} ∤N, then is it true that p∤ \frac{r_{E}}{deg ϕ_{E}}?

Theorem 15. If p is a prime such that p^{2} ∤ N, and p| \frac{r_{E}}{deg ϕ_{E}}, then p|deg ϕ_{E}.

Corollary 16. If p^{2} ∤N and p|r_{E}, then p|deg ϕ_{E}.