CM VALUES AND CENTRAL L-VALUES OF ELLIPTIC MODULAR FORMS
ATSUSHI MURASE
Letf be a Hecke eigenform on SL2(Z) of weightl and Ω an algebraic Hecke character of an imaginary quadratic field K satisfying
(1) Ω((α)) =
α
|α| l
(α∈K×).
The object of this note is to give a formula identifying (up to an ele- mentary constant) the following two invariants attached to (f,Ω) under certain assumption on K:
(i) the central value of the Rankin L-function attached to (f,Ω).
(ii) the square of the “CM-period” of (f,Ω).
Such a formula was first discovered by Waldspurger ([W]) in a very general situation. We will make Waldspurger’s formula more explicit in the full modular case.
To be more precise, let l be an even positive integer and Sl(Γ) the space of holomorphic cusp forms on Γ = SL2(Z) of weight l. We say that f(z) = P∞
m=1af(m) exp(2πimz) ∈ Sl(Γ) is a normalized Hecke eigenform, if af(1) = 1 and if f is a common eigenfunction of all the Hecke operators.
LetK be an imaginary quadratic field of discriminantD, and denote by σ the nontrivial automorphism of K/Q. Let HK be the ideal class group of K, hK = |HK| the class number of K, wK the number of roots of unity in K and ω = ωK/Q the quadratic Dirichlet character corresponding to K/Q. By an ideal of K, we always mean a nonzero fractional ideal ofK. Denote by Na the norm of an ideal aof K.
From now on, we assume that l is divisible by wK. Let Ω be a Hecke character of K satisfying (1). Note that the number of such Ω is hK. Let A ∈ HK and take an ideal a belonging to A. Choosea Z-basis {λ, µ}ofasatisfying Tr(λσµ/√
D) = Na. Then λ−1µ∈H. For f ∈Sl(Γ), the quantity Ω(a) Nal/2λ−lf(λ−1µ) does not depend on the choices of a and {λ, µ}, and we write it for PA(f,Ω). We set
P(f,Ω) = X
A∈HK
PA(f,Ω).
1
2 ATSUSHI MURASE
It is easily seen thatP(f,Ω) is real iffis a normalized Hecke eigenform.
Define a Rankin L-function Z(f,Ω;s) attached to (f,Ω) by Z(f,Ω;s) =L(ω; 2s)X
a
af(Na)Ω(a) Na−(s+(l−1)/2),
where a runs over the integral ideals of K and L(ω;s) is the Dirich- let L-function attached to ω. Note that Z(f,Ω;s) is the convolution of L(f;s) and L(Ω;s). We see that Z∗(f,Ω;s) = (2π)−2s|D|sΓ(s+ 1/2)Γ(s+l−1/2)Z(f,Ω;s) is continued to an entire function ofson the wholeCand satisfies a functional equationZ∗(f,Ω;s) =Z∗(f,Ω; 1−s).
We are now able to state the main results.
Theorem Assume that the class number of K is odd. Let f ∈Sl(Γ) be a normalized Hecke eigenform. Then we have
Z
f,Ω;1 2
= 2l+3πl+1|D|(l−1)/2
(l−1)!wK2 P(f,Ω)2.
Corollary Under the assumptions same as above, the central value Z(f,Ω; 1/2) is real and nonnegative.
Remark
(i) The class number ofK is odd if and only if D=−4,−8 or−p, wherep is an odd prime number with p≡3 (mod 4).
(ii) When hK = 1, we have P(f,Ω) =f((√
D+D)/2).
References
[W] J. L. Waldspurger,Sur les valeurs de certaines fonctionsLautomorphes en leur centre de sym´etrie, Compositio Math.,54(1985), 173–242.
Department of Mathematical Science, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan
E-mail address: [email protected]