Special Values of Modular L-functions

William A. Stein April, 1999 1. Modular Forms

Let k ≥ 2. Attach to a newform

f = ∑ a_{n}q^{n} ∈ S_{k}(Γ_{0}(N), Q)

• M_{f}: Grothendieck motive (Scholl)

• E_{f}: Elliptic curve / R

• L_{f}: Entire function

Invariants.

• Ω_{f} = μ(E_{f}(R)) ∈ R: real volume

• L_{f}(s) ∈ R, s = 1, …, k − 1: special values Are they related?

Theorem. For s = 1, …, k − 1, L_{f}(s) is a rational multiple of (2π)^{s−1}Ω_{f}.

Example: If Δ ∈ S_{12}(SL_{2}(Z)) is Ramanujan’s function then

\frac{L_{Δ}(1)}{Ω_{Δ}} = \frac{2^{2} · 3^{4} · 5 · 7}{691}.

2. Modular Symbols

• V_{k−2} = homogeneous polys of degree k − 2 in X and Y.

• M = Z-module on symbols {α, β}, α, β ∈ P^{1}(Q) modulo

{α, β} + {β, γ} + {γ, α} = 0 • Γ_{0}(N) acts on V_{k−2} ⊗ M:

g(P (X, Y) ⊗ {α, β}) = P (dX − bY, −cX + aY){g(α), g(β)}

• M_{k}(Γ_{0}(N)) = H_{0}(Γ_{0}(N), V_{k−2} ⊗ M)

• B = Z-module on symbols {α}, α ∈ P^{1}(Q).

• B_{k}(Γ_{0}(N)) = H_{0}(Γ_{0}(N), V_{k−2} ⊗ B)

• S_{k}(Γ_{0}(N)) = ker(M_{k}(Γ_{0}(N)) → B_{k}(Γ_{0}(N))), the map is P {α, β} ↦ P {β} − P {α}.

3. Periods M_{k}(Γ_{0}(N)) × S_{k}(Γ_{0}(N), C) → C

⟨P {α, β}, f⟩ ↦ 2πi ∫_{α}^{β} f(z)P (z, 1)dz Period map:

Φ_{f}(x) = ⟨x, f⟩ ∈ C
S_{k}(Γ_{0}(N)) ⟶^{Φ_{f}} C
E_{f} = C/Im(Φ_{f})

Query: What is E_{f}?

E_{f} is an elliptic curve over R. It is possible to numerically approximate j(E_{f}). (Future talk.)

4. Critical Values

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

Critical values are related to periods.

L_{f}(1) = −2πi ∫_{0}^{∞} f(z)dz
= ⟨−Y^{k−2}{0, ∞}, f⟩
L_{f}(2) = −(2π)^{2} ∫_{0}^{∞} f(z)zdz
= 2πi⟨XY^{k−3}{0, ∞}, f⟩
L_{f}(3) = (2π)^{3}i\frac{1}{2!} ∫_{0}^{∞} f(z)z^{2}dz
= \frac{(2π)^{2}}{2!}⟨X^{2}Y^{k−4}\{0, ∞\}, f⟩

So for j an odd integer,

L_{f}(j) = ±\frac{(2π)^{j−1}}{(j − 1)!}⟨X^{j−1}Y^{k−2−(j−1)}\{0, ∞\}, f⟩

5. Rationality Theorem

w_{j} = X^{j−1}Y^{k−2−(j−1)}{0, ∞} ∈ M_{k}(Γ_{0}(N))
Ω_{f}^{0} = μ(A_{f}(R)^{0}) measure of identity component

For embedded Z-modules A, B, let

[A : B] = |det of change of basis matrix|

Theorem.

|\frac{⟨w_{j}, f⟩}{Ω_{f}^{0}}| = [Φ_{f}(S_{k}(Γ_{0}(N)))^{+} : Φ_{f}(Zw_{j})] ∈ Q

Proof.

|⟨w_{j}, f⟩| = |Φ_{f}(w_{j})] = [Z : Φ_{f}(Zw_{j})]

and

|Ω_{f}^{0}| = | min{x > 0 : x ∈ Φ_{f}(S_{k}(Γ_{0}(N)))^{+}|
= [Z : Φ_{f}(S_{k}(Γ_{0}(N)))^{+}|

The index can be computed with Φ_{f} replaced by any map eminating from M_{k}(Γ_{0}(N)) and having the same kernel. Thus the index is rational and

\frac{L_{f}(j)}{(2π)^{j−1} · Ω_{f}} ∈ Q.

6. Examples

Let R(f, s) = \frac{L_{f}(s)}{(2π)^{s−1} · Ω_{f}} · (s − 1)!

\begin{matrix}N & R(f, s), s = 1, …, k − 1, s odd \\ 1 & \frac{2^{2}·3^{4}·5·7}{691}, 7, \frac{3^{2}}{2}, \frac{3^{2}}{2}, 7, \frac{2^{2}·3^{4}·5·7}{691} \\ 3 & \frac{3^{6}·5^{2}}{73}, \frac{3^{3}}{2}, \frac{3}{2}, \frac{1}{2}, \frac{1}{2}, \frac{3·5^{2}}{73} \\ 4 & 2^{5} · 3 · 5^{2}, 2^{6}, 2, \frac{1}{2}, 1, \frac{3·5^{2}}{2^{5}} \\ 5 & \frac{3^{2}·5^{6}·7}{31}, \frac{3^{2}·5^{3}}{2}, \frac{5^{2}}{2}, \frac{5}{2}, \frac{3^{2}}{2}, \frac{3^{2}·5·7}{31} \end{matrix}

Table 1. Rational newforms, k = 12.

k R(f, s), s = 1, …, k − 1, s odd
12 \frac{2^{2}·3^{4}·5·7}{691}, 7, \frac{3^{2}}{2}, \frac{3^{2}}{2}, 7, \frac{2^{2}·3^{4}·5·7}{691}
16 \frac{2^{2}·3^{3}·5·7·11·13}{3617}, 3 · 11, \frac{3·7}{2}, \frac{11}{2}, \frac{11}{2}, \frac{3·7}{2}, 3 · 11, \frac{2^{2}·3^{3}·5·7·11·13}{3617}
18 \frac{2^{5}·3^{3}·5^{4}·7·11·13}{43867}, 2 · 7 · 11 · 13, \frac{3·5^{2}·11}{2}, \frac{3·5·13}{2}, 0, \frac{3·5·13}{2}, \frac{3·5^{2}·11}{2},

2 · 7 · 11 · 13, \frac{2^{5}·3^{3}·5^{4}·7·11·13}{43867}

20 \frac{2^{2}·3^{5}·5·7^{2}·11·13·17}{283·617}, 3 · 11 · 13, \frac{11·13}{2}, \frac{3·11}{2}, \frac{13}{2}, \frac{13}{2}, \frac{3·11}{2}, \frac{11·13}{2},

3 · 11 · 13, \frac{2^{2}·3^{5}·5·7^{2}·11·13·17}{283·617}

22 \frac{2^{4}·3^{3}·5^{4}·7·13·17·19}{131·593}, 2^{4} · 3 · 13 · 17, 3 · 5 · 7 · 13, 13 · 17, \frac{5·17}{2}, 0, \frac{5·17}{2},

13 · 17, 3 · 5 · 7 · 13, 2^{4} · 3 · 13 · 17, \frac{2^{4}·3^{3}·5^{4}·7·13·17·19}{131·593}

26 \frac{2^{4}·3^{4}·5^{3}·7^{2}·11·13·17·19·23}{657931}, 2 · 3 · 11 · 17 · 19 · 43, \frac{5·17·19·97}{2}, \frac{3·11·17·29}{2},

5 · 11 · 19, \frac{17·19}{2}, 0, \frac{17·19}{2}, 5 · 11 · 19, \frac{3·11·17·29}{2}, \frac{5·17·19·97}{2}, 2 · 3 · 11 · 17 · 19 · 43, \frac{2^{4}·3^{4}·5^{3}·7^{2}·11·13·17·19·23}{657931}

Table 2. Level 1 rational newforms.