On the Rational Part of the L-function Attached to a Modular Abelian Variety

William Stein  Amod Agashe

May 1999

Abstract We give formulas for the rational part of the special values of the Lfunction attached to a modular abelian variety. We obtain a result about the denominators of these rational numbers which is consistent with the Birch and Swinnerton-Dyer conjecture.

1 Introduction

Without question one of the most fascinating and important constellation of problems in number theory springs forth from the conjectures of Birch and Swinnerton-Dyer, which relate the special value of an L-function to a host of arithmetic invariants an elliptic curve E:

L(E, 1) = \frac{|� | ∏_{p} c_{p}}{|E(Q)|^{2}} · Ω.

In particular, the value of L(E, s) at 1 is a rational number times a transcendental “period”. Similar conjectures for higher dimensional abelian varieties and even Grothendieck’s motives have sprung to the minds of those such as Tate, and then Beilenson, Deligne, and more recently Bloch and Kato.

As it turns out most (probably all) elliptic curves over Q arise from modular forms by a classical construction of Shimura. If we consider a slightly more general class of newforms than just those spawning elliptic curves we find newforms which give rise to abelian varieties and motives (Scholl). In this paper we study the special values of the L-functions attached to these object and show that they are rational multiples of a certain volume Ω. We give a formula for the rational multiple, show how to compute, and give a bound on the denominator.

This paper follows work of Manin, Mazur, Shimura, and others.

2 The Rational Parts of Special Values

Fix a positive even integer k and a positive integer N. Let f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{0}(N), C) be a newform and let d be the degree of the field Q_{f} = Q(a_{1}, a_{2}, …) generated by the Fourier coefficients. Let S_{f} be the subspace of S_{k}(Γ_{0}(N), C) spanned by the Galois f_{1}, …, f_{d} of f.

2.1 Modular Symbols

Let M = M_{k}(Γ_{0}(N), Z) be the space of weight k modular symbols for Γ_{0}(N), and let S = S_{k}(Γ_{0}(N), Z) be the subspace of symbols “vanishing at the cusps”. More precisely, M is a free abelian group spanned by symbols P (z, y){α, β} with α, β ∈ Q ∪ {∞}, and P ∈ Z[z, y] homogeneous of degree k − 2, modulo the relations

P · ({α, β} + {β, γ} + {γ, α}) = 0

and

P {α, β} = M.P {α, β} = P (dz − by, −cz + ay){M(α), M(β)}

for all M ∈ Γ_{0}(N). The cusp symbol space B is the free abelian group spanned by symbols P {α} subject to the relation P = M.P all M ∈ Γ_{0}(N) Up to sign there is a natural map M → B whose kernel we denote by S. The Hecke operators T_{p} act on M by

T_{p} : P \{α, β\} ↦ [(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) + ∑_{r mod p} (\begin{matrix}1 & r \\ 0 & p \end{matrix})] .P \{α, β\}.

(Omit ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) if p|N.)

2.2 Lattices

Let V be a finite dimensional vector space over R. A lattice L ⊂ V is a subgroup isomorphic to Z^{dim V} such that RL = V. If L, M ⊂ V are lattices define [L : M] to be the absolute value of the determinant of an automorphism of V taking L isomorphically onto M. For convenience, extend the definition to the case when M has rank strictly smaller than dim V by defining [L : M] = 0.

2.3 Duality

There is a pairing ⟨ , ⟩ : S_{k}(Γ_{0}(N), C) × M → C given by

⟨f, z^{j}y^{k−2−j}{α, β}⟩ = ∫_{α}^{β} f(z)z^{j} dz.

We may view the pairing as a map M → Hom(S_{k}, C).

Theorem 2.1. The image of M in Hom(S_{k}, C) is a lattice.

2.4 Shimura’s Abelian Variety

Choose a basis h_{1}, …, h_{d} for S_{f}(Z), the space of cusp forms in S_{f} with rational integer Fourier expansions at ∞. For each h_{i} let h_{i}^{∗} ∈ Hom(S_{f}, C) denote the element defined by h_{i}^{∗}(h_{j}) = δ_{ij}. Let I_{f} : M → Hom(S_{f}, C) be the “period map”:

I_{f}(x) = ∑_{i=1}^{d}⟨h_{i}, x⟩ · h_{i}^{∗}

Theorem 2.2. The image of I_{f} is a lattice. The quotient A_{f} = Hom(S_{f}, C)/I_{f}(S) is an abelian variety defined over R.

2.5 Measures

The choice of basis h_{i}^{∗} for Hom(S_{f}, C) allows us to identify Hom(S_{f}, C) with C^{d} and this defines a measure μ on A_{f}. Lef A_{f}^{±} denote the elements of A_{f} on which complex conjugation acts as ±1. These are also endowed with measures. Let Ω_{f}^{±} be the volume of the identity component of A_{f}^{±}.

2.6 Formula for Special Values

Let L(f, s) = ∑ a_{n}n^{−s} be the Mellin transform of f and set L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s). For j = 1, …, k − 1 let w_{j} = z^{j−1}y^{k−j−1}{0, ∞} ∈ M. For any positive integer n let σ(n) denote the sign of (−1)^{n}.

Theorem 2.3. For j = 1, …, k − 1 the following formula holds:

|L(A_{f}, j)| = \frac{(2π)^{j−1}}{Γ(j)} · Ω_{f}^{σ(j−1)} · [I_{f}(S)^{σ(j−1)} : I_{f}(Tw_{j})]

where the lattice index is taken inside of the R-vector space S_{f}^{σ(j−1)}.

For j = 1, …, k − 1 define

B(A_{f}, j) = \frac{L(A_{f}, j)}{Ω_{f}^{σ(j−1)}} · \frac{Γ(j)}{(2π)^{j−1}} .

Corollary 2.4.

B(A_{f}, j) ∈ \begin{cases}Q & j odd \\ Qi & j even \end{cases}

Let t_{j} be the order of the “rational cuspidal” subgroup of A_{f} generated by the image of I_{f}(w_{j}). It annihilates the denominator of the special value.

Theorem 2.5. For j = 1, 2, … k − 1,

t_{j}B(A_{f}, j) ∈ \begin{cases}Z & j odd \\ Zi & j even \end{cases}

3 Birch and Swinnerton-Dyer Conjecture

Amod’s Section. State BSD. Relate S_{f} to integral basis of Neron differentials.

4 Tables

f  tB/t f  tB/t f  tB/t f  tB/t f  tB/t f  tB/t
11A1 1/5 53B3 2/13 76A1 1/1 96A1  1/2 113D3 2^{3}/7 128D1  1/2
14A1 1/2.3 54A1 1/3 77A1 0/1 96B1  1/2 114A1 1/1 129A1  0/1
15A1 1/2^{2} 54B1 1/3 77B1 1/2 97A3  0/1 114B1 5/2 129B1  3/2
17A1 1/2^{2} 55A1 1/2 77C1 2/3 97B4 2^{3}/2^{3} 114C1 1/1 129C2  1/1
19A1 1/3 55B2 1/2 77D2 1/2 98A1  1/1 115A1 1/1 129D3  2^{2}/11
20A1 1/2.3 56A1 1/2 78A1 1/2 98B2  1/7 115B2 0/1 130A1  0/1
21A1 1/2^{2} 56B1 1/2^{2} 79A1 0/1 99A1  0/1 115C4 2^{2}/2^{3} 130B1  2^{2}/1
23A2 1/11 57A1 0/1 79B5 2/13 99B1  1/1 116A1 1/3 130C1  1/1
24A1 1/2^{2} 57B1 1/2 80A1 1/2 99C1  1/2 116B1 1/2 131A1  0/1
26A1 1/3 57C1 2/5 80B1 1/2 99D1  1/1 116C1 3/1 131B10 2/5.13
26B1 1/7 58A1 0/1 81A2 1/3 100A1 1/1 117A1 0/1 132A1  1/2
27A1 1/3 58B1 2/5 82A1 0/1 101A1 0/1 117B2 1/3 132B1  1/2
29A2 1/7 59A5 1/29 82B2 2/7 101B7 2/5^{2} 117C2 1/1 133A2  0/1
30A1 1/2.3 61A1 0/1 83A1 0/1 102A1 0/1 118A1 0/1 133B2  2^{2}/5
31A2 1/5 61B3 2/5 83B6 2/41 102B1 2/3 118B1 1/1 133C2  0/1
32A1 1/2^{2} 62A1 1/2^{2} 84A1 1/2 102C1 1/1 118C1 2/5 133D3  2^{2}/2^{2}
33A1 1/2 62B2 1/2.3 84B1 1/2 103A2 0/1 118D1 1/1 134A3  1/3
34A1 1/3 63A1 1/2 85A1 1/1 103B6 2^{2}/17 119A4 1/3^{2} 134B3  19/17
35A1 1/3 63B2 1/2.3 85B2 0/1 104A1 1/1 119B5 1/2^{3} 135A1  0/1
35B2 1/2^{3} 64A1 1/2 85C2 2/3 104B2 1/2 120A1 1/2 135B1  2/1
36A1 1/2.3 65A1 0/1 86A2 1/3 105A1 1/2 120B1 1/2 135C2  1/3
37A1 0/1 65B2 1/3 86B2 5/11 105B2 1/2 121A1 0/1 135D2  1/3
37B1 2/3 65C2 1/7 87A2 1/5 106A1 0/1 121B1 1/1 136A1  1/1
38A1 1/3 66A1 1/3 87B3 1/2^{2} 106B1 1/1 121C1 1/1 136B1  0/1
38B1 1/5 66B1 1/2 88A1 0/1 106C1 2^{3}/3 121D1 2/1 136C2  2/2
39A1 1/2 66C1 1/1 88B2 2/2^{2} 106D1 1/3 122A1 0/1 137A4  0/1
39B2 1/2.7 67A1 1/1 89A1 0/1 107A2 0/1 122B2 1/3 137B7 2^{4}/2.17
40A1 1/2 67B2 0/1 89B1 1/2 107B7 2^{2}/53 122C3 2^{3}/31 138A1  0/1
41A3 1/2.5 67C2 2^{2}/11 89C5 2/11 108A1 1/3 123A1 0/1 138B1  1/3
42A1 1/2^{2} 68A2 1/2.3 90A1 1/3 109A1 1/1 123B1 0/1 138C1  1/2
43A1 0/1 69A1 1/2 90B1 1/3 109B3 0/1 123C2 2/7 138D2  2/2
43B2 2/7 69B2 1/2 90C1 1/1 109C4 2^{3}/3^{2} 123D3 2/2^{2} 139A1  1/1
44A1 1/3 70A1 1/2 91A1 0/1 110A1 1/3 124A1 1/1 139B3  0/1
45A1 1/2 71A3 1/5 91B1 0/1 110B1 1/3 124B1 0/1 139C7  2^{3}/23
46A1 1/2 71B3 1/7 91C2 2/7 110C1 1/1 125A2 0/1 140A1  1/1
47A4 1/23 72A1 1/2 91D3 2/2^{3} 110D2 1/2.3 125B2 2^{2}/5 140B1  1/1
48A1 1/2 73A1 1/2 92A1 1/3 111A3 1/2 125C4 1/5 141A1  0/1
49A1 1/2 73B2 0/1 92B1 0/1 111B4 7/2.19 126A1 1/1 141B1  1/2
50A1 1/3 73C2 2^{2}/3 93A2 0/1 112A1 0/1 126B1 1/1 141C1  1/2^{2}
50B1 1/5 74A2 1/3 93B3 2^{2}/2^{4} 112B1 1/2 127A3 0/1 141D1  2/1
51A1 1/3 74B2 5/19 94A1 1/2 112C1 1/1 127B7 2^{3}/3.7 141E1  0/1
51B2 1/2^{2} 75A1 1/1 94B2 1/2 113A1 1/2 128A1 0/1 141F2  2/2
52A1 1/2 75B1 1/2 95A3 1/2.5 113B2 1/2 128B1 1/2 142A1  0/1
53A1 0/1 75C1 1/5 95B4 1/2.3 113C3 0/1 128C1 1/2 142B1  1/2

Figure 1: Special Values for S_{2}(Γ_{0}(N))

f  tB/t  f  tB/t  f  tB/t  f  tB/t  f  tB/t
5A1  \frac{2.5}{13}, \frac{1}{1}  16A1  \frac{2^{3}}{1} , \frac{1}{1}  23B4 \frac{2.23^{4}}{2.53} , \frac{23}{1} 29B5 \frac{2^{5}.29^{5}}{421} , \frac{2^{2}.29^{2}}{1} 34C2 \frac{2^{3}.3.17^{2}}{3} , \frac{2.3^{2}}{1}
6A1  \frac{2.3}{5}, \frac{1}{1}  17A1  \frac{17}{2.3}, \frac{0}{1}  24A1 \frac{2.3}{1}, \frac{2}{1} 30A1  \frac{2.3.5}{1}, \frac{1}{1}  35A1  \frac{5.7}{2}, \frac{0}{1}
7A1  \frac{7}{2.5}, \frac{1}{1}  17B3 \frac{2^{2}.17^{3}}{29} , \frac{2.17}{1} 25A1  \frac{5^{2}}{1} , \frac{2}{1}  30B1  \frac{2^{2}.3.5}{1}, \frac{5}{1}  35B2  \frac{2.5^{2}.7^{2}}{1} , \frac{7}{1}
8A1  \frac{2}{1}, \frac{1}{1}  18A1  \frac{2.3}{1}, \frac{1}{1}  25B1 \frac{2^{2}.5}{1}, \frac{2.5}{1} 31A2  \frac{2.31^{2}}{2^{2}} , \frac{0}{1}  35C3  \frac{2.5^{3}.7^{3}}{2^{3}.3} , \frac{2.5}{1}
9A1  \frac{2}{1}, \frac{1}{1}  19A1  \frac{19}{3}, \frac{0}{1}  25C1  \frac{5}{1}, \frac{0}{1}  31B5 \frac{2.3^{2}.31^{5}}{2.13.37} , \frac{2.31^{2}}{1} 36A1  \frac{2^{2}.3^{2}}{1} , \frac{1}{1}
10A1  \frac{2.5}{3}, \frac{1}{1}  19B3 \frac{2.3.19^{3}}{181} , \frac{2.19}{1} 26A1 \frac{2^{2}.13}{5}, \frac{1}{1} 32A1  \frac{2^{3}}{1} , \frac{1}{1}  37A4  \frac{2^{3}.37^{4}}{3.19} , \frac{0}{1}
11A2 \frac{2.11^{2}}{61} , \frac{1}{1} 20A1  \frac{5}{1}, \frac{1}{1}  26B1 \frac{2^{2}.13}{1}, \frac{1}{1} 32B1  \frac{2^{3}}{1} , \frac{2}{1}  37B5 \frac{2^{7}.3.37^{5}}{137} , \frac{2^{4}.37^{3}}{1}
12A1  \frac{3}{1}, \frac{1}{1}  21A1  \frac{2.3.7}{5}, \frac{7}{1}  26C1 \frac{2.13}{3}, \frac{11}{1} 32C1  \frac{2^{3}}{1} , \frac{0}{1}  38A1  \frac{2.19}{5}, \frac{0}{1}
13A1  \frac{0}{7}, \frac{0}{1}  21B1  \frac{3.7}{2}, \frac{0}{1}  27A1  \frac{3^{2}}{1} , \frac{2}{1}  33A1  \frac{3.11}{5}, \frac{0}{1}  38B2  \frac{2^{3}.19^{2}}{1} , \frac{19}{1}
13B2 \frac{2.13^{2}}{17} , \frac{2.13}{1} 21C2 \frac{2.3^{2}.7^{2}}{2^{3}} , \frac{2}{1} 27B1  \frac{3^{2}}{1} , \frac{0}{1}  33B1  \frac{3.11}{1}, \frac{0}{1}  38C2  \frac{2^{3}.19^{2}}{3} , \frac{2.3^{2}}{1}
14A1  \frac{2.7}{5}, \frac{1}{1}  22A1  \frac{2.11}{1}, \frac{11}{1}  27C2 \frac{2.3^{4}}{1} , \frac{3}{1} 33C2 \frac{2.3^{2}.11^{2}}{1} , \frac{2.3.11}{1} 39A1  \frac{2.3.13}{1}, \frac{0}{1}
14B1  \frac{2^{2}.7}{3}, \frac{1}{1}  22B1  \frac{2.11}{5}, \frac{0}{1}  28A1 \frac{2.7}{1}, \frac{0}{1} 33D2 \frac{2.3^{2}.11^{2}}{2^{2}} , \frac{2.11}{1} 39B2  \frac{3^{2}.13^{2}}{2.5} , \frac{13}{1}
15A1  \frac{3.5}{2}, \frac{1}{1}  22C1  \frac{2.11}{3}, \frac{2}{1}  28B1  \frac{7}{1}, \frac{7}{1}  34A1  \frac{2^{2}.17}{5}, \frac{17}{1}  39C3 \frac{2^{2}.3^{4}.13^{3}}{2^{3}} , \frac{19}{1}

Figure 2: Special Values for S_{4}(Γ_{0}(N)) at s = 1, 2

f  tB/t  f  tB/t  f  tB/t
3A1  \frac{2.3^{3}}{13} , \frac{3}{1}, \frac{1}{1}  13B3 \frac{2^{4}.3^{2}.5.13^{7}}{61} , \frac{2^{2}.3.13^{3}}{1} , \frac{2^{3}.13^{3}}{1} 19A1  \frac{2^{3}.3.19^{2}}{5.7} , \frac{19^{2}}{1} , \frac{0}{1}
4A1  \frac{2.3}{1}, \frac{2^{2}}{1} , \frac{1}{1}  14A1  \frac{2^{3}.7^{2}}{3^{2}} , \frac{2.7}{1}, \frac{1}{1}  19B1  \frac{2.7.19^{2}}{3} , \frac{3.19}{1}, \frac{19}{1}
5A1  \frac{2.5^{3}}{31} , \frac{5}{1}, \frac{1}{1}  14B1  \frac{2^{3}.7^{2}}{1} , \frac{2.7}{1}, \frac{11}{1}  19C2  \frac{2^{2}.19^{4}}{7} , \frac{2^{2}.19^{2}}{1} , \frac{0}{1}
6A1  \frac{2^{3}.3^{2}}{7} , \frac{2.3}{1}, \frac{1}{1}  15A1  \frac{2^{3}.3^{2}.5^{2}}{7} , \frac{3.5}{1}, \frac{0}{1}  19D4 \frac{2^{4}.3^{4}.11.19^{9}}{127} , \frac{2.3.19^{5}}{1} , \frac{2^{3}.19^{4}}{1}
7A1  \frac{2^{3}.7^{2}}{43} , \frac{7}{1}, \frac{0}{1}  15B1  \frac{3^{2}.5^{2}}{2} , \frac{3.5}{1}, \frac{3}{1}  20A1  \frac{2^{4}.5^{2}}{1} , \frac{2^{2}.5}{1}, \frac{2.3}{1}
7B2  \frac{2.3^{2}.7^{5}}{2.19} , \frac{7^{2}}{1} , \frac{7^{2}}{1}  15C2  \frac{2.3^{5}.5^{4}}{2^{2}} , \frac{3^{3}.5^{2}}{1} , \frac{5^{2}}{1}  21A1  \frac{3^{2}.7^{2}}{2^{2}} , \frac{2.3.7}{1}, \frac{0}{1}
8A1  \frac{2^{5}}{1} , \frac{2^{3}}{1} , \frac{2}{1}  16A1  \frac{2^{8}}{1} , \frac{2^{4}}{1} , \frac{0}{1}  21B1  \frac{2^{2}.3^{2}.7^{2}}{1} , \frac{3.7}{1}, \frac{2.7}{1}
9A1  \frac{2^{2}.3^{2}}{1} , \frac{2.3^{2}}{1} , \frac{2}{1}  16B1  \frac{2^{8}.3}{1}, \frac{2^{4}.3}{1}, \frac{2^{4}}{1}  21C1  \frac{3^{3}.7^{2}}{2} , \frac{2^{2}.3^{2}.7}{1}, \frac{2.5}{1}
10A1  \frac{2^{3}.5^{2}}{3^{2}} , \frac{2.5}{1}, \frac{0}{1}  17A1  \frac{3.17^{2}}{1} , \frac{2.17}{1}, \frac{0}{1}  21D1  \frac{2.3^{2}.7^{2}}{1} , \frac{3.7}{1}, \frac{3.5}{1}
10B1  \frac{2^{3}.3.5^{2}}{1} , \frac{2.5}{1}, \frac{5^{2}}{1}  17B1  \frac{2^{3}.17^{2}}{3.13} , \frac{3.17}{1}, \frac{0}{1}  22A1  \frac{2^{3}.11^{2}}{3^{2}} , \frac{2.11}{1}, \frac{0}{1}
10C1  \frac{2^{3}.3.5^{2}}{7} , \frac{2.5}{1}, \frac{2^{2}}{1}  17C4 \frac{2^{6}.3.5^{2}.17^{9}}{2.307} , \frac{2^{6}.3.17^{4}}{1} , \frac{2^{4}.17^{3}}{1} 22B1  \frac{2^{4}.3.11^{2}}{1} , \frac{2.11}{1}, \frac{2^{2}.11}{1}
11A1  \frac{2^{2}.3.11^{2}}{37} , \frac{11}{1}, \frac{0}{1}  18A1  \frac{2^{3}.3^{2}}{1} , \frac{2.3^{2}}{1} , \frac{0}{1}  22C1  \frac{2^{4}.11^{2}}{7} , \frac{2.11^{2}}{1} , \frac{0}{1}
11B3 \frac{2^{2}.3.11^{7}}{5.19} , \frac{3.11^{3}}{1} , \frac{11^{2}}{1} 18B1  \frac{2^{4}.3^{2}}{1} , \frac{2.3^{2}}{1} , \frac{2}{1}  22D2 \frac{2^{6}.3.7.11^{4}}{1} , \frac{2^{2}.3.11^{3}}{1} , \frac{2^{5}.11^{2}}{1}
13A2  \frac{2^{3}.13^{4}}{157} , \frac{2.13^{2}}{1} , \frac{0}{1}  18C1  \frac{2^{3}.3^{4}}{1} , \frac{2.3^{2}}{1} , \frac{2^{2}.3}{1}  23A3  \frac{2^{8}.3^{2}.23^{6}}{13^{2}} , \frac{2.3.23^{4}}{1} , \frac{0}{1}

Figure 3: Special Values for S_{6}(Γ_{0}(N)) at s = 1, 2, 3

f  tB/t
1A1  \frac{2^{3}.3^{4}.5.7}{691}, \frac{2^{4}.3}{1}, \frac{2.7}{1}, \frac{5^{2}}{1} , \frac{3^{2}}{1} , \frac{2^{2}.5}{1}
3A1  \frac{2.3^{6}.5^{2}}{73} , \frac{3^{6}}{1} , \frac{3^{3}}{1} , \frac{2.3^{3}}{1} , \frac{3}{1}, \frac{2.5}{1}
4A1  \frac{2^{6}.3.5^{2}}{1} , \frac{2^{9}}{1} , \frac{2^{7}}{1} , \frac{2^{4}}{1} , \frac{2^{2}}{1} , \frac{0}{1}
5A1  \frac{2.3^{2}.5^{6}.7}{31}, \frac{3.5^{5}}{1} , \frac{3^{2}.5^{3}}{1} , \frac{2^{3}.5^{2}}{1} , \frac{5^{2}}{1} , \frac{0}{1}
5B2 \frac{2^{5}.3^{4}.5^{12}.7}{601}, \frac{2^{2}.3^{3}.5^{9}.7}{1}, \frac{2^{2}.5^{7}.7}{1}, \frac{2.3.5^{6}.7}{1}, \frac{2^{2}.3.5^{3}}{1} , \frac{2.5^{4}}{1}
6A1  \frac{2^{8}.3^{6}.7}{5}, \frac{2^{5}.3^{4}.7}{1}, \frac{2^{4}.3^{3}.7}{1}, \frac{2^{2}.3^{2}.7}{1}, \frac{2.3^{3}}{1} , \frac{3^{2}}{1}
6B1  \frac{2^{5}.3^{6}.5^{2}.7}{13}, \frac{2^{5}.3^{4}}{1} , \frac{2^{3}.3^{3}.17}{1}, \frac{2^{2}.3^{2}}{1} , \frac{2.3^{3}}{1} , \frac{0}{1}
6C1  \frac{2^{5}.3^{6}.5}{7}, \frac{2^{5}.3^{5}.5}{1}, \frac{2^{3}.3^{3}}{1} , \frac{2^{2}.3^{2}.19}{1}, \frac{2.3}{1}, \frac{2^{3}.5}{1}

Figure 4: Special Values for S_{12}(Γ_{0}(N)) at s = 1, 2, 3, 4, 5, 6 Figure 5: Table of Abelian varieties of level N ≤ 1000 such that the odd part of the analytic � is > 1.