Preprint (July 17, 1999), Version 0.1

The BSD Conjecture for J_{0}(65)

W. A. Stein

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Introduction

The Jacobian J = J_{0}(65) of the modular curve X_{0}(65) is a 5 dimensional abelian variety whose Mordell-Weil group has rank 1. It is isogenous over Q to a product A × B × C where A is the elliptic curve y^{2} =, and B is the Jacobian of the curve curve, and C is the Jacobian of curve. It is the modular Jacobian of lowest level for which the Birch and Swinnerton-Dyer conjecture (BSD) predicts that the Shafarevich-Tate group is nontrivial [Verify this! It might VERY WELL be false!!]

Let L(J, s) be the canonical L-series of J. The Birch and Swinnerton-Dyer conjecture asserts that

L′(J, 1) =^{?} \frac{|� | · c_{5} · c_{13}}{T^{2}} · R · Ω_{J}

where � is the Shafarevich-Tate group, T is the torsion subgroup of J(Q), c_{5} and c_{13} are the number of components of the special fiber of the Néron model of J, R is the regulator and Ω_{J} is the measure of J(R) with respect to a basis of Néron differentials. In this paper we compute each of the quantities c_{5}, c_{13}, T, R, L′(J, 1) and a quantity which differs from Ω_{J} by a Manin constant which is probably equal to 1.

The goal of this short note is to compute |� |, assuming the truth of the BSD conjecture.

Contents

1 Some equations 2

2 Computing the invariants 2 2.1 Leading coefficient at s = 1 . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Real volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5 Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5.1 Detailed computation of c_{5} . . . . . . . . . . . . . . . . . . . . 4 2.5.2 Detailed computation of c_{13} . . . . . . . . . . . . . . . . . . . . 5

3 Analytic order of the Shafarevich-Tate group 6

1 Some equations

To be concrete, we give equations for the abelian varieties under consideration, though we will not make use of these equations. According to [Gal96] the curve X_{0}(65) is cut out by the following equations:

v^{2} − y^{2} + 2x^{2} + 2z^{2} = 0
v^{2} − 2y^{2} + z^{2} + w^{2} − 2wx = 0
y^{2} + 2yz + 2z^{2} − w^{2} − x^{2} = 0

According to [Cre97, FpFS^{+}99], the 3 optimal new factors A, B, and C are all Jacobians of curves:

A = Jac(y^{2} + xy = x^{3} − x)
B = Jac(y^{2} = −x^{6} + 10x^{5} − 32x^{4} + 20x^{3} + 40x^{2} + 6x − 1
C = Jac(y^{2} + (x^{3} + 1)y = −4x^{6} + 9x^{4} + 7x^{3} + 18x^{2} − 10)

2 Computing the invariants

A lucky fact is that J_{0}(65) is new in the sense that X_{0}(5) and X_{0}(13) have genus 0. This makes it reasonable to expect that the following assumption holds:

Assumption: The Manin constant is 1, so that the real volume Ω_{J} is equal to the volume computed using a basis of Néron differentials.

2.1 Leading coefficient at s = 1

The canonical L-function associated to an abelian variety is invariant under isogeny. In particular, since J ∼ A × B × C we have L(J, s) = L(A × B × C, s) = L(A, s)L(B, s)L(C, s). Since B and C have rank 0 and A has rank 1, the leading coefficient of L(J, s) as s = 1 is the derivative L′(J, s)/1! = L′(J, s). By the product rule we have

L′(J, s) = L′(A, s)L(B, s)L(C, s)
+L(A, s)L′(B, s)L(C, s) + L(A, s)L(B, s)L′(C, s).

Using the that L(A, 1) = 0, taking L′(A, 1) from [Cre97], and computing L(B, 1), L(C, 1) using HECKE [Ste99] (500 terms of q-expansion), we obtain

L′(J, 1) = L′(A, 1)L(B, 1)L(C, 1)
≈ 0.50533434230686 · 0.91225087803795579498 · 0.45206783187309768436
≈ 0.2083995171877410261569391894

2.2 Torsion

I guess that |J(Q)_{tor}| = 168 = 2^{3} ·3·7. The upper bound coming from counting points on J(F_{p}) using Hecke operators is 2^{3} · 3 · 7. Each of A, B, and C possess exactly one nontrivial rational 2-torsion point. We also have |A(Q)_{tor}| = 2, |B(Q)_{tor}| = 6, and |C(Q)_{tor} = 14, so 2 · 3 · 7 is a lower bound.

After computing all other invariants (see below) we find that the only possible value for J(Q)_{tor} so that the order of � predicted by BSD is an integer is |J(Q)_{tor}| = 168.

2.3 Regulator

There is an injective map A ↪ J and the image generates the J(Q), up to torsion. I GUESS that Reg(J) = Reg(A). We find that

Reg(A) ≈ 0.3755140986612663218044728765

(using PARI’s ellheightmatrix(e,[[1,0]]), then double checking in Cremona’s tables.)

WARNING: I have not proved that Reg(J) = Reg(A).

2.4 Real volume

The volume with respect to an integral basis at infinity is

Ω_{J} ≈ 93.23516483763867.

This was computed using HECKE.

2.5 Tamagawa numbers

We have

\overline{c}_{5} = 42, \overline{c}_{13} = 6.

These values were computed both by the character group method (see below) and via the Edixhoven-Mazur-Raynaud formula. Using a more careful analysis of character groups, we find that c_{5} = 14 and c_{13} = 6. The details are given below.

2.5.1 Detailed computation of c_{5}

Let D be the free abelian group generated by the supseringular points on the modular curve X_{0}(13)/F_{5}. The character group X_{5}(J) of the toric part of X_{0}(65)/F_{5} is the subgroup of D of elements of degree 0 (the degree is the sum of the coefficients on the canonical basis of superingular points).

Proposition 2.1. c_{5} = 14

Some data about the character group X_{5}(J).

Monodromy weights: w_{1} = w_{2} = 3, w_{3} = w_{4} = w_{5} = w_{6} = 1.

Operators: The matrix of T_{5} = Frob on the group of divisors on the supersingular points:

Frob_{5} = (\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix})

Presentation of the component group Φ_{5}(J): Choose the following basis for the group X = X_{5}(J) of divisors of degree 0:

b_{1} = e_{1} − e_{2}, b_{2} = e_{1} − e_{3}, b_{3} = e_{1} − e_{4}, b_{4} = e_{1} − e_{5}, b_{5} = e_{1} − e_{6}.

The component group Φ fits into the sequence

0 ⟶ X ⟶ Hom(X , Z) ⟶ Φ ⟶ 0.

Thus Φ is the quotient of the group b_{1}^{∗}, …, b_{5}^{∗} by the subgroup generated by

\{ \begin{matrix}6b_{1}^{∗} + 3b_{2}^{∗} + 3b_{3}^{∗} + 3b_{4}^{∗} + 3b_{5}^{∗} \\ 3b_{1}^{∗} + 4b_{2}^{∗} + 3b_{3}^{∗} + 3b_{4}^{∗} + 3b_{5}^{∗} \\ 3b_{1}^{∗} + 3b_{2}^{∗} + 4b_{3}^{∗} + 3b_{4}^{∗} + 3b_{5}^{∗} \\ 3b_{3}^{∗} + 3b_{2}^{∗} + 3b_{3}^{∗} + 4b_{4}^{∗} + 3b_{5}^{∗} \\ 3b_{3}^{∗} + 3b_{2}^{∗} + 3b_{3}^{∗} + 3b_{4}^{∗} + 4b_{5}^{∗} \end{matrix} \}

Letting [x] denote a class in the quotient group, we find

[b_{2}^{∗}] = [b_{3}^{∗}] = [b_{4}^{∗}] = [b_{5}^{∗}] = 3[b_{1}^{∗}]

and that 42[b_{1}^{∗}] = 0.

To compute the subgroup fixed by Frob = T_{p} we must find the action of Frob on Hom(X , Z). The monodromy pairing ⟨ , ⟩ : X × X → Z is T-invariant, and thus we can find the T-module Hom(X , Z) inside of X ⊗Q. It is the set of elements x ∈ X ⊗Q such that ⟨x, y⟩ ∈ Z for all y ∈ X. The dual basis b_{1}^{∗}, b_{2}^{∗}, b_{3}^{∗}, b_{4}^{∗}, b_{5}^{∗} ∈ X ⊗ Q is a set of elements so that b_{i}^{∗}(b_{j}) = δ_{ij} (Kronecker’s δ). For convenience, we will work instead in D ⊗ Q. If v ∈ D ⊗ Q corresponds to b_{1}^{∗} then we have the matrix equation

(\begin{matrix}1 & −1 & 0 & 0 & 0 & 0 \\ 1 & 0 & −1 & 0 & 0 & 0 \\ 1 & 0 & 0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 & −1 & 0 \\ 1 & 0 & 0 & 0 & 0 & −1 \\ 1 & 1 & 3 & 3 & 3 & 3 \end{matrix}) · (\begin{matrix}3 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) · v = (\begin{matrix}1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}) .

Letting a denote the first matrix, and b the second diagonal one, we find that the action of Frob on Hom(X, Z) with respect to the basis b_{1}^{∗}, b_{2}^{∗}, b_{3}^{∗}, b_{4}^{∗}, b_{5}^{∗}, Eis^{∗} is given by

a · b · Frob ·(a · b)^{−1} = (\begin{matrix}−1 & 0 & 0 & 0 & 0 & 0 \\ −1 & 0 & 1 & 0 & 0 & 0 \\ −1 & 1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 1 & 0 \\ −1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) .

Therefore

Frob(b_{1}^{∗}) = −(b_{1}^{∗} + b_{2}^{∗} + b_{3}^{∗} + b_{4}^{∗} + b_{5}^{∗}) = −(13b_{1}^{∗}).

Multiplication by −13 on a cyclic group of order 42 has exactly 14 fixed points, so c_{5} = 14.

2.5.2 Detailed computation of c_{13}

Proposition 2.2. c_{13} = 6

Some data about the character group X_{13}(J).

Monodromy weights: w_{1} = w_{2} = w_{3} = w_{4} = w_{5} = w_{6} = 1.

Operators: The matrix of T_{13} = Frob on the group of divisors on the supersingular points:

Frob_{13} = (\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix})

Presentation of the group: Choose the following basis for the group X = X_{13}(J) of divisors of degree 0:

b_{1} = e_{1} − e_{2}, b_{2} = e_{1} − e_{3}, b_{3} = e_{1} − e_{4}, b_{4} = e_{1} − e_{5}, b_{5} = e_{1} − e_{6}.

The component group Φ fits into the sequence

0 ⟶ X ⟶ Hom(X , Z) ⟶ Φ ⟶ 0.

Since the monodromy weights are trivial, Φ is the quotient of the group b_{1}, …, b_{5} by the subgroup generated by

\{ \begin{matrix}2b_{1}^{∗} + b_{2}^{∗} + b_{3}^{∗} + b_{4}^{∗} + b_{5}^{∗}, \\ b_{1}^{∗} + 2b_{2}^{∗} + b_{3}^{∗} + b_{4}^{∗} + b_{5}^{∗}, \\ b_{1}^{∗} + b_{2}^{∗} + 2b_{3}^{∗} + b_{4}^{∗} + b_{5}^{∗}, \\ b_{1}^{∗} + b_{2}^{∗} + b_{3}^{∗} + 2b_{4}^{∗} + b_{5}^{∗}, \\ b_{1}^{∗} + b_{2}^{∗} + b_{3}^{∗} + b_{4}^{∗} + 2b_{5}^{∗}. \end{matrix} \}

Backsubstituting, we find

[b_{1}^{∗}] = [b_{2}^{∗}]n = [b_{3}^{∗}] = [b_{4}^{∗}] = [b_{5}^{∗}]

and that 6[b_{1}^{∗}] = 0.

Just as before we compute

a · b · Frob ·(a · b)^{−1} == (\begin{matrix}−1 & 0 & 0 & 0 & 0 & 0 \\ −1 & 0 & 1 & 0 & 0 & 0 \\ −1 & 1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 1 & 0 \\ −1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) .

Therefore

Frob(b_{1}^{∗}) = −(b_{1}^{∗} + b_{2}^{∗} + b_{3}^{∗} + b_{4}^{∗} + b_{5}^{∗}) = −(5b_{1}^{∗}).

Since −5 ≡ 1 (mod 6) the action of Frobenius is trivial so c_{13} = 6.

3 Analytic order of the Shafarevich-Tate group

The BSD conjecture predicts that

L′(J, 1) =^{?} \frac{|� | · c_{5} · c_{13}}{|J(Q)_{tor}|^{2}} · Reg ·Ω_{J}

Assuming this and solving for |� | we obtain

|� | = \frac{L′(J, 1)}{Reg ·Ω_{J}} · \frac{J(Q)_{tor}^{2}}{c_{5} · c_{13}}
= \frac{0.2083995171877410261569391894}{0.3755140986612663218044728765 · 93.23516483763867} · \frac{168^{2}}{14 · 6}
≈ \frac{1}{168} · \frac{168^{2}}{14 · 6}
≈ 2.

References

[Cre97] J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[FpFS^{+}99] E. V. Flynn, Leprévost F., E. F. Schaeffer, M. Stoll, and Wetherell J. L., Empirical evidence for the birch and swinnerton-dyer conjectures for modular jacobians of genus 2 curves, Preprint (1999).

[Gal96] S. D. Galbraith, Equations for modular curves, Oxford Ph.D. thesis (1996), 91 pp.

[Ste99] W. A. Stein, HECKE: The modular forms calculator, Software (available online) (1999).