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Modular Forms

Kenneth A. Ribet  William A. Stein

December 5, 2003

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Contents

1 L-functions 1

1.1 L-functions Attached to Modular Forms . . . . . . . . . . . . . . . 1 1.1.1 Analytic Continuation and Functional Equation . . . . . . 2 1.1.2 A Conjecture About Nonvanishing of L(f, k/2) . . . . . . . 3 1.1.3 Euler Products . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.4 Visualizing L-function . . . . . . . . . . . . . . . . . . . . . 5

2 The Birch and Swinnerton-Dyer Conjecture 7 2.1 The Rank Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Refined Rank Zero Conjecture . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The Number of Real Components . . . . . . . . . . . . . . 11 2.2.2 The Manin Index . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 The Real Volume Ω_{A} . . . . . . . . . . . . . . . . . . . . . . 13 2.2.4 The Period Mapping . . . . . . . . . . . . . . . . . . . . . . 13 2.2.5 Manin-Drinfeld Theorem . . . . . . . . . . . . . . . . . . . 14 2.2.6 The Period Lattice . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.7 The Special Value L(A, 1) . . . . . . . . . . . . . . . . . . . 14 2.2.8 Rationality of L(A, 1)/Ω_{A} . . . . . . . . . . . . . . . . . . . 15

2.3 General Refined Conjecture . . . . . . . . . . . . . . . . . . . . . . 17 2.4 The Conjecture for Non-Modular Abelian Varieties . . . . . . . . . 18 2.5 Numerical Evidence for the Conjectures . . . . . . . . . . . . . . . 19

References 21

6 Contents

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2

The Birch and Swinnerton-Dyer Conjecture

This chapter is about the conjecture of Birch and Swinnerton-Dyer on the arithmetic of abelian varieties. We focus primarily on abelian varieties attached to modular forms.

In the 1960s, Sir Peter Swinnerton-Dyer worked with the EDSAC computer lab at Cambridge University, and developed an operating system that ran on that computer (so he told me once). He and Bryan Birch programmed EDSAC to compute various quantities associated to elliptic curves. They then formulated the conjectures in this chapter in the case of dimension 1 (see [Bir65, Bir71, SD67]). Tate formulated the conjectures in a functorial way for abelian varieties of arbitrary dimension over global fields in [Tat66], and proved that if the conjecture is true for an abelian variety A, then it is also true for each abelian variety isogenous to A.

Suitably interpreted, the conjectures may by viewed as generalizing the analytic class number formula, and Bloch and Kato generalized the conjectures to Grothendieck motives in [BK90].

2.1 The Rank Conjecture

Let A be an abelian variety over a number field K.

Definition 2.1.1 (Mordell-Weil Group). The Mordell-Weil group of A is the abelian group AK) of all K-rational points on A.

Theorem 2.1.2 (Mordell-Weil). The Mordell-Weil group A(K) of A is finitely generated.

The proof is nontrivial and combines two ideas. First, one proves the “weak Mordell-Weil theorem”: for any integer m the quotient A(K)/mA(K) is finite. This is proved by combining Galois cohomology techniques with standard finiteness theorems from algebraic number theory. The second idea is to introduce the Néron-Tate canonical height h : A(K) → R_{≥0} and use properties of h to deduce, from finiteness of A(K)/mA(K), that A(K) itself is finitely generated.

Definition 2.1.3 (Rank). By the structure theorem A(K) ≅ Z^{r} ⊕ G_{tor}, where r is a nonnegative integer and G_{tor} is the torsion subgroup of G. The rank of A is r.

Let f ∈ S_{2}(Γ_{1}(N)) be a newform of level N, and let A = A_{f} ⊂ J_{1}(N) be the corresponding abelian variety. Let f_{1}, …, f_{d} denote the Gal(\overline{Q}/Q)-conjugates of f, so if f = ∑ a_{n}q^{n}, then f_{i} = ∑ σ(a_{n})q^{n}, for some σ ∈ Gal(\overline{Q}/Q).

Definition 2.1.4 (L-function of A). We define the L-function of A = A_{f} (or any abelian variety isogenous to A) to be

L(A, s) = ∏_{i=1}^{d} L(f_{i}, s).

By Theorem 1.1.4, each L(f_{i}, s) is an entire function on C, so L(A, s) is entire. In Section 2.4 we will discuss an intrinsic way to define L(A, s) that does not require that A be attached to a modular form. However, in general we do not know that L(A, s) is entire.

Conjecture 2.1.5 (Birch and Swinnerton-Dyer). The rank of A(Q) is equal to ord_{s=1} L(A, s).

One motivation for Conjecture 2.1.5 is the following formal observation. Assume for simplicity of notation that dim A = 1. By Theorem 1.1.6, the L-function L(A, s) = L(f, s) has an Euler product representation

L(A, s) = ∏_{p|N} \frac{1}{1 − a_{p}p^{−s}} · ∏_{p∤N} \frac{1}{1 − a_{p}p^{−s} + p · p^{−2s}} ,

which is valid for Re(s) sufficiently large. (Note that ε = 1, since A is a modular elliptic curve, hence a quotient of X_{0}(N).) There is no loss in considering the product L^{∗}(A, s) over only the good primes p ∤ N, since ord_{s=1} L(A, s) = ord_{s=1} L^{∗}(A, s) (because ∏_{p|N} \frac{1}{1−a_{p}p^{−s}} is nonzero at s = 1). We then have formally that

L^{∗}(A, 1) = ∏_{p∤N} \frac{1}{1 − a_{p}p^{−1} + p^{−1}}

= ∏_{p∤N} \frac{p}{p − a_{p} + 1}

= ∏_{p∤N} \frac{p}{#A(F_{p})}

The intuition is that if the rank of A is large, i.e., A(Q) is large, then each group A(F_{p}) will also be large since it has many points coming from reducing the elements of A(Q) modulo p. It seems likely that if the groups #A(F_{p}) are unusually large, then L^{∗}(A, 1) = 0, and computational evidence suggests the more precise Conjecture 2.1.5. Example 2.1.6. Let A_{0} be the elliptic curve y^{2} +y = x^{3} −x^{2}, which has rank 0 and conductor 11, let A_{1} be the elliptic curve y^{2} + y = x^{3} − x, which has rank 1 and conductor 37, let A_{2} be the elliptic curve y^{2} + y = x^{3} + x^{2} − 2x, which has rank 2 and conductor 389, and finally let A_{3} be the elliptic curve y^{2} + y = x^{3} − 7x + 6, which has rank 3 and conductor 5077. By an exhaustive search, these are known to be the smallest-conductor elliptic curves of each rank. Conjecture 2.1.5 is known to be true for them, the most difficult being A_{3}, which relies on the results of [GZ86].

The following diagram illustrates |#A_{i}(F_{p})| for p < 100, for each of these curves. The height of the red line (first) above the prime p is |#A_{0}(F_{p})|, the green line (second) gives the value for A_{1}, the blue line (third) for A_{2}, and the black line (fourth) for A_{3}. The intuition described above suggests that the clumps should look like triangles, with the first line shorter than the second, the second shorter than the third, and the third shorter than the fourth—however, this is visibly not the case. The large Mordell-Weil group over Q does not increase the size of every E(F_{p}) as much as we might at first suspect. Nonetheless, the first line is no longer than the last line for every p except p = 41, 79, 83, 97.

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Remark 2.1.7. Suppose that L(A, 1) ≠ 0. Then assuming the Riemann hypothesis for L(A, s) (i.e., that L(A, s) ≠ 0 for Re(s) > 1), Goldfeld [Gol82] proved that the Euler product for L(A, s), formally evaluated at 1, converges but does not converge to L(A, 1). Instead, it converges (very slowly) to L(A, 1)/\sqrt{2}. For further details and insight into this strange behavior, see [Con03].

Remark 2.1.8. The Clay Math Institute has offered a one million dollar prize for a proof of Conjecture 2.1.5 for elliptic curves over Q. See [Wil00].

Theorem 2.1.9 (Kolyvagin-Logachev). Suppose f ∈ S_{2}(Γ_{0}(N)) is a newform such that ord_{s=1} L(f, s) ≤ 1. Then Conjecture 2.1.5 is true for A_{f}.

10 2. The Birch and Swinnerton-Dyer Conjecture

Theorem 2.1.10 (Kato). Suppose f ∈ S_{2}(Γ_{1}(N)) and L(f, 1) ≠ 0. Then Conjecture 2.1.5 is true for A_{f}.

2.2 Refined Rank Zero Conjecture

Let f ∈ S_{2}(Γ_{1}(N)) be a newform of level N, and let A_{f} ⊂ J_{1}(N) be the corresponding abelian variety.

The following conjecture refines Conjecture 2.1.5 in the case L(A, 1) ≠ 0. We recall some of the notation below, where we give a formula for L(A, 1)/Ω_{A}, which can be computed up to an vinteger, which we call the Manin index. Note that the definitions, results, and proofs in this section are all true exactly as stated with X_{1}(N) replaced by X_{0}(N), which is relevant if one wants to do computations.

Conjecture 2.2.1 (Birch and Swinnerton-Dyer). Suppose L(A, 1) ≠ 0. Then

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

By Theorem 2.1.10, the group � (A) is finite, so the right hand side makes sense. The right hand side is a rational number, so if Conjecture 2.2.1 is true, then the quotient L(A, 1)/Ω_{A} should also be a rational number. In fact, this is true, as we will prove below (see Theorem 2.2.11). Below we will discuss aspects of the proof of rationality in the case that A is an elliptic curve, and at the end of this section we give a proof of the general case.

In to more easily understanding L(A, 1)/Ω_{A}, it will be easiest to work with A = A_{f}^{∨}, where A_{f}^{∨} is the dual of A_{f}. We view A naturally as a quotient of J_{1}(N) as follows. Dualizing the map A_{f} ↪ J_{1}(N) we obtain a surjective map J_{1}(N) → A_{f}^{∨}. Passing to the dual doesn’t affect whether or not L(A, 1)/Ω_{A} is rational, since changing A by an isogeny does not change L(A, 1), and only changes Ω_{A} by multiplication by a nonzero rational number.

2.2.1 The Number of Real Components

Definition 2.2.2 (Real Components). Let c_{∞} be the number of connected components of A(R).

If A is an elliptic curve, then c_{∞} = 1 or 2, depending on whether the graph of the affine part of A(R) in the plane R^{2} is connected. For example, Figure 2.2.1 shows the real points of the elliptic curve defined by y^{2} = x^{3} − x in the three affine patches that cover P^{2}. The completed curve has two real components.

In general, there is a simple formula for c_{∞} in terms of the action of complex conjugation on H_{1}(A(R), Z), which can be computed using modular symbols. The formula is

log_{2}(c_{∞}) = dim_{F_{2}} A(R)[2] − dim(A).

2.2.2 The Manin Index

The map J_{1}(N) → A induces a map J → A on Néron models. Pullback of differentials defines a map

H^{0}(A, Ω_{A/Z}^{1}) → H^{0}(J , Ω_{J /Z}^{1}).  (2.2.1)

One can show that there is a q-expansion map

H^{0}(J , Ω_{J /Z}^{1}) → Z[[q]]  (2.2.2)

y  x  ^{z  x}  ^{y  z}

FIGURE 2.2.1. Graphs of real solutions to y^{2}z = x^{3} − xz^{2} on three affine patches

which agrees with the usual q-expansion map after tensoring with C. (For us X_{1}(N) is the curve that parameterizes pairs (E, μ_{N} ↪ E), so that there is a q-expansion map with values in Z[[q]].)

Let φ_{A} be the composition of (2.2.1) with (2.2.2).

Definition 2.2.3 (Manin Index). The Manin index c_{A} of A is the index of φ_{A}(H^{0}(A, Ω_{A/Z}^{1})) in its saturation. I.e., it is the order of the quotient group

(\frac{Z[[q]]}{φ_{A}(H^{0}(A, Ω_{A/Z}^{1}))})_{tor} .

Open Problem 2.2.4. Find an algorithm to compute c_{A}.

Manin conjectured that c_{A} = 1 when dim A = 1, and I think c_{A} = 1 in general.

Conjecture 2.2.5 (Agashe, Stein). c_{A} = 1.

This conjecture is false if A is not required to be attached to a newform, even if A_{f} ⊂ J_{1}(N)^{new}. For example, Adam Joyce, a student of Kevin Buzzard, found an A ⊂ J_{1}(431) (and also A′ ⊂ J_{0}(431)) whose Manin constant is 2. Here A is isogenous over Q to a product of two elliptic curves. Also, the Manin index for J_{0}(33) (viewed as a quotient of J_{0}(33)) is divisible by 3, because there is a cusp form in S_{2}(Γ_{0}(33)) that has integer Fourier expansion at ∞, but not at one of the other cusps.

Theorem 2.2.6. If f ∈ S_{2}(Γ_{0}(N)) then the Manin index c of A_{f}^{∨} can only divisible by 2 or primes whose square divides N. Moreover, if 4 ∤ N, then ord_{2}(c) ≤ dim(A_{f}).

The proof involves applying nontrivial theorems of Raynaud about exactness of sequences of differentials, then using a trick with the Atkin-Lehner involution, which was introduced by Mazur in [Maz78], and finally one applies the “q-expansion principle” in characteristic p to deduce the result (see [AS]). Also, Edixhoven claims he can prove that if A_{f} is an elliptic curve then c_{A} is only divisible by 2, 3, 5, or 7. His argument use his semistable models for X_{0}(p^{2}), but my understanding is that the details are not all written up.

2.2.3 The Real Volume Ω_{A}

Definition 2.2.7 (Real Volume). The real volume Ω_{A} of A(R) is the volume of A(R) with respect to a measure obtained by wedging together a basis for H^{0}(A, Ω^{1}).

If A is an elliptic curve with minimal Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

then one can show that

ω = \frac{dx}{2y + a_{1}x + a_{3}}  (2.2.3)

is a basis for H^{0}(A, Ω^{1}). Thus

Ω_{A} = ∫_{A(R)} \frac{dx}{2y + a_{1}x + a_{3}} .

There is a fast algorithm for computing Ω_{A}, for A an elliptic curve, which relies on the quickly-convergent Gauss arithmetic-geometric mean (see [Cre97, §3.7]). For example, if A is the curve defined by y^{2} = x^{3} − x (this is a minimal model), then

Ω_{A} ∼ 2 × 2.622057554292119810464839589.

For a general abelian variety A, it is an open problem to compute Ω_{A}. However, we can compute Ω_{A}/c_{A}, where c_{A} is the Manin index of A, by explicitly computing A as a complex torus using the period mapping Φ, which we define in the next section.

2.2.4 The Period Mapping

Let

Φ : H_{1}(X_{1}(N), Z) → Hom_{C}(Cf_{1} + ⋯ + Cf_{d}, C)

be the period mapping on integral homology induced by integrating homology classes on X_{0}(N) against the C-vector space spanned by the Gal(\overline{Q}/Q)-conjugates f_{i} of f. Extend Φ to H_{1}(X_{1}(N), Q) by Q-linearity. We normalize Φ so that Φ({0, ∞})(f) = L(f, 1). More explicitly, for α, β ∈ P^{1}(Q), we have

Φ({α, β})(f) = −2πi ∫_{α}^{β} f(z)dz.

The motivation for this normalization is that

L(f, 1) = −2πi ∫_{0}^{i∞} f(z)dz,  (2.2.4)

which we see immediately from the Mellin transform definition of L(f, s):

L(f, s) = (2π)^{s}Γ(s)^{−1} ∫_{0}^{i∞}(−iz)^{s}f(z)\frac{dz}{z} .

2.2.5 Manin-Drinfeld Theorem

Recall the Manin-Drinfeld theorem, which we proved long ago, asserts that {0, ∞} ∈ H_{1}(X_{0}(N), Q). We proved this by explicitly computing (p + 1 − T_{p})({0, ∞}), for p ∤ N, noting that the result is in H_{1}(X_{0}(N), Z), and inverting p + 1 − T_{p}. Thus there is an integer n such that n{0, ∞} ∈ H_{1}(X_{0}(N), Z).

Suppose that A = A_{f}^{∨} is an elliptic curve quotient of J_{0}(N). Rewriting (2.2.4) in terms of Φ, we have Φ({0, ∞}) = L(f, 1). Let ω be a minimal differential on A, as in (2.2.3), so ω = −c_{A} · 2πif(z)dz, where c_{A} is the Manin index of A, and the equality is after pulling ω back to H^{0}(X_{0}(N), Ω) ≅ S_{2}(Γ_{0}(N)). Note that when we defined c_{A}, there was no factor of 2πi, since we compared ω with f(q)\frac{dq}{q}, and q = e^{2πiz}, so dq/q = 2πidz.

2.2.6 The Period Lattice

The period lattice of A with respect to a nonzero differential g on A is

L_{g} = {∫_{γ} g : γ ∈ H_{1}(A, Z)} ,

and we have A(C) ≅ C/L_{g}. This is the Abel-Jacobi theorem, and the significance of g is that we are choosing a basis for the one-dimensional C-vector space Hom(H^{0}(A, Ω), C), in order to embed the image of H_{1}(A, Z) in C.

The integral ∫_{A(R)} g is “visible” in terms of the complex torus representation of A(C) = C/L_{g}. More precisely, if L_{g} is not rectangular, then A(R) may be identified with the part of the real line in a fundamental domain for L_{g}, and ∫_{A(R)} g is the length of this segment of the real line. If L_{g} is rectangular, then it is that line along with another line above it that is midway to the top of the fundamental domain.

The real volume, which appears in Conjecture 2.2.1, is

Ω_{A} = ∫_{A(R)} ω = −c_{A} · 2πi ∫_{A(R)} f.

Thus Ω_{A} is the least positive real number in L_{ω} = −c_{A} · 2πiL_{f}, when the period lattice is not rectangular, and twice the least positive real number when it is.

2.2.7 The Special Value L(A, 1)

Proposition 2.2.8. We have L(f, 1) ∈ R.

Proof. With the right setup, this would follow immediately from the fact that z ↦ −\overline{z} fixes the homology class \{0, ∞\}. However, we don’t have such a setup, so we give a direct proof.

Just as in the proof of the functional equation for Λ(f, s), use that f is an eigenvector for the Atkin-Lehner operator W_{N} and (2.2.4) to write L(f, 1) as the sum of two integrals from i/\sqrt{N} to i∞. Then use the calculation

\overline{2πi ∫_{i/\sqrt{N}}^{i∞} ∑_{n=1}^{∞} a_{n}e^{2πinz}dz} = −2πi ∑_{n=1}^{∞} a_{n}\overline{∫_{i/\sqrt{N}}^{i∞} e^{2πinz}dz}

= −2πi ∑_{n=1}^{∞} a_{n} \overline{\frac{1}{2πin}e^{−2πn/\sqrt{N}}}

= 2πi ∑_{n=1}^{∞} a_{n} \frac{1}{2πin}e^{2πn/\sqrt{N}}

to see that \overline{L(f, 1)} = L(f, 1).

Remark 2.2.9. The BSD conjecture implies that L(f, 1) ≥ 0, but this is unknown (it follows from GRH for L(f, s)).

2.2.8 Rationality of L(A, 1)/Ω_{A}

Proposition 2.2.10. Suppose A = A_{f} is an elliptic curve. Then L(A, 1)/Ω_{A} ∈ Q. More precisely, if n is the smallest multiple of {0, ∞} that lies in H_{1}(X_{0}(N), Z) and c_{A} is the Manin constant of A, then 2n · c_{A} · L(A, 1)/Ω_{A} ∈ Z.

Proof. By the Manin-Drinfeld theorem n{0, ∞} ∈ H_{1}(X_{0}(N), Z), so

n · L(f, 1) = −n · 2πi · ∫_{0}^{i∞} f(z)dz ∈ −2πi · L_{f} = \frac{1}{c_{A}} L_{ω}.

Combining this with Proposition 2.2.8, we see that

n · c_{A} · L(f, 1) ∈ L_{ω}^{+},

where L_{ω}^{+} is the submodule fixed by complex conjugation (i.e., L_{ω}^{+} = L∩R). When the period lattice is not rectangular, Ω_{A} generates L_{ω}^{+}, and when it is rectangular, \frac{1}{2}Ω_{A} generates. Thus n · c_{A} · L(f, 1) is an integer multiple of \frac{1}{2}Ω_{A}, which proves the proposition.

Proposition 2.2.10 can be more precise and generalized to abelian varieties A = A_{f}^{∨} attached to newforms. One can also replace n by the order of the image of (0) − (∞) in A(Q).

Theorem 2.2.11 (Agashe, Stein). Suppose f ∈ S_{2}(Γ_{1}(N)) is a newform and let A = A_{f}^{∨} be the abelian variety attached to f. Then we have the following equality of rational numbers:

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

Note that L(A, 1) ∈ R, so |L(A, 1)| = ±L(A, 1), and one expects, of course, that L(A, 1) ≥ 0.

For V and W lattices in an R-vector space M, the lattice index [V : W] is by definition the absolute value of the determinant of a change of basis taking a basis for V to a basis for W, or 0 if W has rank smaller than the dimension of M.

Proof. Let Ω̃_{A} be the measure of A(R) with respect to a basis for S_{2}(Γ_{1}(N), Z)[I_{f}], where I_{f} is the annihilator in T of f. Note that Ω̃_{A} · c_{A} = Ω_{A}, where c_{A} is the Manin index. Unwinding the definitions, we find that

Ω̃_{A} = c_{∞} · [Hom(S_{2}(Γ_{1}(N), Z)[I_{f}], Z) : Φ(H_{1}(X_{0}(N), Z))^{+}].

For any ring R the pairing

T_{R} × S_{2}(Γ_{1}(N), R) → R

given by ⟨T_{n}, f⟩ = a_{1}(T_{n}f) is perfect, so (T/I_{f}) ⊗ R ≅ Hom(S_{2}(Γ_{1}(N), R)[I_{f}], R). Using this pairing, we may view Φ as a map

Φ : H_{1}(X_{1}(N), Q) → (T/I_{f}) ⊗ C,

so that

Ω̃_{A} = c_{∞} · [T/I_{f} : Φ(H_{1}(X_{0}(N), Z))^{+}].

Note that (T/I_{f}) ⊗ C is isomorphic as a ring to a product of copies of C, with one copy corresponding to each Galois conjugate f_{i} of f. Let π_{i} ∈ (T/I_{f}) ⊗ C be the projector onto the subspace of (T/I_{f}) ⊗ C corresponding to f_{i}. Then

Φ({0, ∞}) · π_{i} = L(f_{i}, 1) · π_{i}.

Since the π_{i} form a basis for the complex vector space (T/I_{f}) ⊗ C, if we view Φ({0, ∞}) as the operator “left-multiplication by Φ({0, ∞})”, then

det(Φ({0, ∞})) = ∏_{i} L(f_{i}, 1) = L(A, 1),

Letting H = H_{1}(X_{0}(N), Z), we have

[Φ(H)^{+} : Φ(T{0, ∞})] = [Φ(H)^{+} : (T/I_{f}) · Φ({0, ∞})]

= [Φ(H)^{+} : T/I_{f}] · [T/I_{f} : T/I_{f} · Φ({0, ∞})]

= \frac{c_{∞}}{Ω̃_{A}} · | det(Φ(\{0, ∞\}))|

= \frac{c_{∞}c_{A}}{Ω_{A}} · |L(A, 1)|,

which proves the theorem.

Remark 2.2.12. Theorem 2.2.11 is false, in general, when A is a quotient of J_{1}(N) not attached to a single Gal(\overline{Q}/Q)-orbit of newforms. It could be modified to handle this more general case, but the generalization seems not to has been written down.

2.3 General Refined Conjecture 17

2.3 General Refined Conjecture

Conjecture 2.3.1 (Birch and Swinnerton-Dyer). Let r = ord_{s=1} L(A, s). Then r is the rank of A(Q), the group � (A) is finite, and

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

2.4 The Conjecture for Non-Modular Abelian Varieties

Conjecture 2.3.1 can be extended to general abelian varieties over global fields. Here we discuss only the case of a general abelian variety A over Q. We follow the discussion in [Lan91, 95-94] (Lang, Number Theory III), which describes Gross’s formulation of the conjecture for abelian varieties over number fields, and to which we refer the reader for more details.

For each prime number ℓ, the ℓ-adic Tate module associated to A is

Ta_{ℓ}(A) = \underleftarrow{lim}_{n} A(\overline{Q})[ℓ^{n}].

Since A(\overline{Q})[ℓ^{n}] ≅ (Z/ℓ^{n}Z)^{2 dim(A)}, we see that Ta_{ℓ}(A) is free of rank 2 dim(A) as a Z_{ℓ}-module. Also, since the group structure on A is defined over Q, Ta_{ℓ}(A) comes equipped with an action of Gal(\overline{Q}/Q):

ρ_{A,ℓ} : Gal(\overline{Q}/Q) → Aut(Ta_{ℓ}(A)) ≈ GL_{2d}(Z_{ℓ}).

Suppose p is a prime and let ℓ ≠ p be another prime. Fix any embedding \overline{Q} ↪ \overline{Q}_{p}, and notice that restriction defines a homorphism r : Gal(\overline{Q}_{p}/Q_{p}) → Gal(\overline{Q}/Q). Let G_{p} ⊂ Gal(\overline{Q}/Q) be the image of r. The inertia group I_{p} ⊂ G_{p} is the kernel of the natural surjective reduction map, and we have an exact sequence

0 → I_{p} → Gal(\overline{Q}_{p}/Q_{p}) → Gal(\overline{F}_{p}/F_{p}) → 0.

The Galois group Gal(\overline{F}_{p}/F_{p}) is isomorphic to Ẑ with canonical generator x ↦ x^{p}. Lifting this generator, we obtain an element Frob_{p} ∈ Gal(\overline{Q}_{p}/Q_{p}), which is welldefined up to an element of I_{p}. Viewed as an element of G_{p} ⊂ Gal(\overline{Q}/Q), the element Frob_{p} is well-defined up I_{p} and our choice of embedding \overline{Q} ↪ \overline{Q}_{p}. One can show that this implies that Frob_{p} ∈ Gal(\overline{Q}/Q) is well-defined up to I_{p} and conjugation by an element of Gal(\overline{Q}/Q).

For a G_{p}-module M, let

M^{I_{p}} = {x ∈ M : σ(x) = x all σ ∈ I_{p}}.

Because I_{p} acts trivially on M^{I_{p}}, the action of the element Frob_{p} ∈ Gal(\overline{Q}/Q) on M^{I_{p}} is well-defined up to conjugation (I_{p} acts trivially, so the “up to I_{p}” obstruction vanishes). Thus the characteristic polynomial of Frob_{p} on M^{I_{p}} is welldefined, which is why L_{p}(A, s) is well-defined. The local L-factor of L(A, s) at p is

L_{p}(A, s) = \frac{1}{det (I − p^{−s} Frob_{p}^{−1} | Hom_{Z_{ℓ}}(Ta_{ℓ}(A), Z_{ℓ})^{I_{p}})}.

Definition 2.4.1. L(A, s) = ∏_{all p} L_{p}(A, s)

For all but finitely many primes Ta_{ℓ}(A)^{I_{p}} = Ta_{ℓ}(A). For example, if A = A_{f} is attached to a newform f = ∑ a_{n}q^{n} of level N and p ∤ ℓ·N, then Ta_{ℓ}(A)^{I_{p}} = Ta_{ℓ}(A). In this case, the Eichler-Shimura relation implies that L_{p}(A, s) equals ∏ L_{p}(f_{i}, s), where the f_{i} = ∑ a_{n,i}q^{n} are the Galois conjugates of f and L_{p}(f_{i}, s) = (1 − a_{p,i} · p^{−s} + p^{1−2s})^{−1}. The point is that Eichler-Shimura can be used to show that the characteristic polynomial of Frob_{p} is ∏_{i=1}^{dim(A)}(X^{2} − a_{p,i}X + p^{1−2s}).

Theorem 2.4.2. L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s).

2.5 Numerical Evidence for the Conjectures 19

2.5 Numerical Evidence for the Conjectures

20 2. The Birch and Swinnerton-Dyer Conjecture

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