The Denominator of the Special Value L(A_{f}, 1)/Ω(A_{f})

Amod Agashe and William Stein^{∗}

March 11, 1999

Abstract

Let f be a newform and A_{f} the quotient of J_{0}(N) constructed by Shimura. We prove that, up to a Manin constant and a power of 2, the denominator of the rational number L(A_{f}, 1)/Ω(A_{f}) divides the order of the image of (0) − (∞) in A_{f}(Q). This provides evidence for the Birch and Swinnerton-Dyer conjecture and raises questions about the structure of A_{f}(Q).

1 Introduction

Fix a positive integer N. Let f ∈ S_{2}(Γ_{0}(N)) be a newform and let A_{f} be the corresponding optimal quotient of J_{0}(N). The L-function of A_{f} is

L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s)

where f_{1}, …, f_{d} are the Galois conjugates of f. Let

Ω(A_{f}) = ∫_{A_{f} (R)} |ω|

∗email: amod and was at math.berkeley.edu.

where ω is a differential d-form on the Neron model of A_{f}. For p|N the analogous local quantity is c_{p} which is the number of F_{p}-rational components of the special fiber of the Neron model of A_{f} at p. Let

� (A_{f}) = Ker[H^{1}(Q, A_{f}) → ∏_{v} H^{1}(Q_{v}, A_{f})]

where the product is over all primes p and ∞.

The Birch and Swinnerton-Dyer conjecture (BSD conjecture), as generalized by Tate, predicts that � (A_{f}) is finite and

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

When #A_{f}(Q) is infinite the right hand side is 0. This conjecture is currently the subject of much intensive research. We do have

Theorem 1.1 (Kolyvagin, Logachev). If L(A_{f}, 1) ≠ 0 then both A_{f}(Q) and � (A_{f}) are finite.

We first express L(A_{f}, 1) in terms of modular symbols in order to show that the denominator divides the order of a certain subgroup of #A_{f}(Q). Not only does our result provide evidence for the BSD conjecture, but assuming the BSD conjecture it also suggests that the natural map

A_{f}(Q) → ∏_{p|N} Φ_{A_{f} ,p}

should be (very close to) injective.

2 Modular Symbols Expression for L(A_{f}, 1)

Fix a newform f as in the introduction and assume that L(A_{f}, 1) ≠ 0.

Let H_{1}(X_{0}(N), Z) be the first integral homology of the modular curve X_{0}(N). The Hecke algebra T and the involution ∗ both act on H_{1}(X_{0}(N), Z), and their actions commute. We have an exact sequence

H_{1}(X_{0}(N), Z) ⟶^{Φ_{f}} C^{d} → A_{f}(C) → 0

where

Φ_{f}(γ) = (∫_{γ} f_{1}, …, ∫_{γ} f_{d}).

Let e = {0, ∞} ∈ H_{1}(X_{0}(N), Q) correspond to integration along the vertical path from 0 to i∞.

Definition 2.1 (Module Index). Let V be a Q-vector space and let L, M ⊂ V be lattices (of full rank). Denote by [L : M] the absolute value of the determinant of an automorphism of V sending L to M.

Define

L(f) = [Φ_{f}(H_{1}(X_{0}(N), Z)^{+}) : Φ_{f}(Te)]

where the images are lattices in V = Φ_{f}(H_{1}(X_{0}(N), Q)^{+}).

Theorem 2.2. We have

L(f) = \frac{|L(A_{f}, 1)|}{Ω(A_{f})} · c_{∞} · c_{M}

where c_{∞} is the number of real components and c_{M} is the Manin constant.

3 The Denominator of the Special Value

Fix a newform f = ∑ a_{n}q^{n} as in the introduction and assume

L(A_{f}, 1) ≠ 0.

The cusps 0 and ∞ on X_{0}(N) give rise to a rational torsion point

P_{e} = (0) − (∞) ∈ J_{0}(N).

Let \overline{P}_{e} denote the image of P_{e} in A_{f}(Q) and let

C_{e} = Z\overline{P}_{e}

be the cyclic subgroup of A_{f}(Q) which it generates. Let

𝔭_{f} = {t ∈ T : t(f) = 0}.

The Hecke algebra T acts on A_{f} through a quotient:

O_{f} = T/𝔭_{f} ≅ Z[a_{1}, a_{2}, …].

Let π : T → O_{f} denote the natural surjection.

Proposition 3.1. The group C_{e} ⊂ A_{f}(Q) is O_{f}-invariant.

Proof. It suffices to show that each π(T_{p}) acts as a scalar on \overline{P}_{e}, since the π(T_{p}) generate O_{f}. Let p be a prime. Case p ∤ N: Then, following the proof the Manin-Drinfeld theorem,

T_{p}P_{e} = (p + 1)P_{e}.

Thus

π(T_{p})\overline{P}_{e} = (p + 1)\overline{P}_{e}.

Case p ∤ N: Then π(T_{p}) = a_{p} ∈ {0, ±1} (see bottom of page 64 of [DI]). Thus

π(T_{p})\overline{P}_{e} ∈ \{0, ±\overline{P}_{e}\} ⊂ C_{e}.

Theorem 3.2. The denominator of L(f) divides the order of the cyclic subgroup C_{e} ⊂ A_{f}(Q).

Proof. In the definition of L(f) we can identify V = Φ_{f}(H_{1}(X_{0}(N), Q)^{+}) with the Q-vector space H_{1}(X_{0}(N), Q)^{+}/ Ker(Φ_{f}). Thus we may replace Φ_{f} by any homomorphism Φ eminating from H_{1}(X_{0}(N), Z) and having the same kernel as Φ_{f}. The resulting module index L(f) remains unchanged. Choose some nonzero

Φ ∈ Hom(H_{1}(X_{0}(N), Z)^{+}, O_{f})

satisfying the following additional requirement:

Φ(tγ) = π(t)Φ(γ), for all t ∈ T

By “multiplicity one” and duality between homology and differentials, Φ exists and is uniquely determined up to a nonzero scalar in O_{f}.

Both Φ(H_{1}(X_{0}(N), Z)^{+}) and Φ(Te) are contained in K = Frac(O_{f}). They are fractional O_{f}-ideals. Furthermore

L(f) = [Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(Te)].

Next define an ideal I ⊂ O_{f} by exactness of

0 → I → O_{f} ⟶^{t↦t.\overline{P}_{e}} C_{e} → 0.

The map O_{f} → C_{e} is surjective because O_{f} contains Z and C_{f} is cyclic as an abelian group. Thus O_{f}/I ≅ C_{e} is finite cyclic. Furthermore, the Abel-Jacobi theorem implies that

I = {t ∈ O : tΦ(e) ∈ Φ(H_{1}(X_{0}(N), Z)^{+})}.

We are now in a position to bound the denominator of L(f). Writing H = H_{1}(X_{0}(N), Z)^{+} we have

L(f) = [Φ(H) : Φ(Te)] = [Φ(H) : O_{f}Φ(e)]
= [Φ(H) : IΦ(e)] · [IΦ(e) : O_{f}Φ(e)]
= \frac{[Φ(H) : IΦ(e)]}{[O_{f}Φ(e) : IΦ(e)]}.

Next observe that:

1) IΦ(e) ⊂ Φ(H) because of the construction of I, and

2) IΦ(e) ⊂ O_{f}Φ(e) because I ⊂ O_{f}. It follows that

[Φ(H) : IΦ(e)] ∈ Z and [O_{f}Φ(e) : IΦ(e)] ∈ Z.

Thus the denominator of L(f) divides

[O_{f}Φ(e) : IΦ(e)] = [O_{f} : I] = #(O_{f}/I) = #C_{e}.

Question 3.3. Note that the ideal class of the ideal

Φ(H_{1}(X_{0}(N), Z)^{+}) ⊂ O_{f}

is independent of the choice of Φ. What is the significance of this class in the ideal class group of O_{f}? What is its order?

Question 3.4. In the proof we expressed L(f) as a quotient

L(f) = \frac{[Φ(H) : IΦ(e)]}{[O_{f}Φ(e) : IΦ(e)]}.

Both the numerator and the denominator are well-defined, irregardless of the choice of Φ. How do they relate to the numerator and the denominator in the BSD conjecture? In particular, can we:

connect [Φ(H) : IΦ(e)] with #� (A_{f}) · | ∏_{p|N} c_{p}|, or

connect [O_{f}Φ(e) : IΦ(e)] with #A_{f}(Q) · #A_{f}^{∨}(Q)?

4 Idea to Bound A_{f}(Q)

Fix a newform f of level N and assume that L(A_{f}, 1) ≠ 0. Theorem 4.1. Suppose p ∤ 2N. Then

A_{f}(Q) ↪ A_{f}(F_{p}).

Thus #A_{f}(Q) divides

G(f) = gcd{#A_{f}(F_{p}) : p ∤ 2N}.

Let the notation be as in the proof of Theorem 3.2. Thus we have

O_{f} = T/𝔭_{f} = Z[a_{1}, a_{2}, …]

and a map

Φ : H_{1}(X_{0}(N), Z)^{+} → O_{f}

such that

Φ(tγ) = π(t)Φ(γ).

We also have the ideal

I = {t ∈ O_{f} : tΦ(e) ∈ Φ(H_{1}(X_{0}(N), Z)^{+})}.

When proving the theorem we observed that

#(O/I) = #C_{e} | #A_{f}(Q).

Consider the ideal L ⊂ O_{f} generated by the “obvious” elements of I:

L = ((p + 1) − π(T_{p}) : p ∤ 2N) ⊂ I

We will use the following theorem to relate #(O_{f}/L) to G(f). Theorem 4.2. Suppose p ∤ N. Let F (x) be the characteristic polynomial of π(T_{p}). Then

#A_{f}(F_{p}) = F (p + 1).

Define the norm of x ∈ O_{f} to be the determinant of the linear map ℓ_{x} = left multiplication by x. Observe that

| Norm(x)| = [O_{f} : xO_{f}] = #(O_{f}/xO_{f}).

Corollary 4.3. Suppose p ∤ N. Then

Norm((p + 1) − π(T_{p})) = #A_{f}(F_{p}).

Proof. If F (x) is the characteristic polynomial of π(T_{p}) then F (p + 1) is the determinant of left multiplication by (p + 1) − π(T_{p}).

Lemma 4.4. Let 𝔞 be an ideal of O_{f}. Then

#(O_{f}/𝔞) | gcd{Norm(x) : x ∈ 𝔞}.

Proof. We have

#(O_{f}/𝔞) | [O_{f} : 𝔞] · [𝔞 : xO_{f}] = [O_{f} : xO_{f}] = | Norm(x)| ∈ Z.

Thus

#(O_{f}/L) | G(f) = gcd{#A_{f}(F_{p}) : p ∤ 2N}.

Remark 4.5. When I first looked at this I thought that maybe there would be equality. (For example, if O_{f} = Z there is indeed equality.) But, I see no reason for equality now.

In the elliptic case the index of L in I measures the failure of C_{e} to equal E(Q).

Proposition 4.6. Suppose E = A_{f} has dimension one. Then

#(O_{f}/I) | #E(Q) | #(O_{f}/L).

In particular

#(E(Q)/C_{e}) | [I : L].

Proof. L is the ideal in O_{f} = Z generated by the elements p + 1 − a_{p} = #A_{f}(F_{p}) for p ∤ 2N.

Question 4.7. To what extent does this observation carry over to higher dimensional A_{f}? What is the relationship between A_{f}(Q) and C_{e}? Is #(A_{f}(Q)/C_{e}) a power of 2?

5 Numerical Data

Let G′(f) = gcd{#A_{f}(F_{p}) : p ∤ 2N, p ≤ 97}.

\begin{matrix}f & L(f) & |C_{e}| & |A_{f}(Q)| & G′(f) \\ 11A1 & 1/5 & 5 & 5 & 5 \\ 35B2 & 1/8 & 8 & ? & 16 \end{matrix}

[This table will be extended later.]

References

[DI] F. Diamon, J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem, CMS Conference Proceedings, Volume 17, (1994).