Values of $L$-series of modular forms at the center of the critical strip.
In this paper, the authors give an elementary proof of the
following theorem. Notations: Let
f(z)={sum}{sub}(n >= 1) a(n) exp(2i pi nz),
a(1)=1, be a cusp form for SL(2, Z) of weight 2k, which is an eigenfunction
of all the Hecke operators. Shimura's theory of forms of half integral weight
associates to f a cusp form g of weight k+1/2, which is an eigenform of all
the Hecke operators: g(z)={sum}{sub}(n >= 1) c(n) exp(2i pi nz). The first
author [Math. Ann. 248 (1980), no. 3, 249 - 266; MR 81j:10030] has shown that
one can suppose c(n)=0 unless ( - 1){sup}kn{iden}0 or 1 (mod 4). The Fourier
coefficients of f and g are related by
c(n{sup}2{vert}D{vert})=c({vert}D{vert}){sum}{lim}d{vert}n d > 0{term}
mu (d)((D)/(d))d{sup}(k - 1)a((n)/(d)),
where D is an arbitrary fundamental discriminant (i.e. 1 or the discriminant
of a quadratic field), with ( - 1){sup}kD > 0, (D/d) the Kronecker symbol, mu
(d) the Mobius function. Let L(f, D, s) be the "twisted" L-series of f,
defined by analytic continuation of the Dirichlet series {sum}{sub}(n >=
1)(D/n)a(n)n{sup}( - s). Let us denote by {lelbo}f, f{relbo} and {lelbo}g, g
{relbo} the Petersson scalar products:
{lelbo}g, g{relbo}=(1/6){int}{lim}H/ Gamma {sub}0(4){term}{vert}g(z)
{vert}{sup}2y{sup}(k - 3/2) dx dy,
{lelbo}f, f{relbo}={int}{lim}H/SL(2,Z){term}{vert}f(z){vert}{sup}2y
{sup}(2k - 2) dx dy,
where H is the upper half-plane, and "6" enters as the index of Gamma
{sub}0(4)=(({sup}a{sub}c {sup}b{sub}d){in}SL(2, Z): 4{vert}c) in SL(2, Z).
Theorem: With the above notations,
(c({vert}D{vert}){sup}2)/({lelbo}g, g{relbo})=((k - 1)!)/(pi {sup}k)
{vert}D{vert}{sup}(k - 1/2)(L(f, D, k))/({lelbo}f, f{relbo}).
The sign of c({vert}D{vert}) is still mysterious. The form g, which is
determined up to a scalar multiple, can be chosen in such a way that its
Fourier coefficients c(n) lie in the field Q{sub}f generated by the a(n) over
Q. A remarkable fact is then the following: there exists a complex number c
{neq}0, associated to f, such that the value L(f, D, k) is an algebraic
multiple of c. The algebraic number is a perfect square of Q{sub}f, up to a
trivial factor.
This result is compatible with the Birch - Swinnerton-Dyer conjecture, for
the value L(f, 1) of a cusp form attached to an elliptic curve E/Q.
The question of describing the numbers c(n) (n square free) and in
particular of relating them to critical values of L(f, s) was mentioned by G.
Shimura in the "Miscellaneous remarks" of his article [Ann. of Math. (2) 97
(1973), 440 - 481; MR 48 #10989]. It was brilliantly answered by J.-L.
Waldspurger [J. Math. Pures Appl (9) 59 (1980), no. 1, 1 - 133; ibid., to
appear], in the general case where f is a cusp form of even weight 2k >= 2
for a congruence subgroup. The above theorem is a version of Waldspurger's
theorem in the special case of SL(2, Z). However, Waldspurger's proof does
not give the value of the constant of proportionality between the L(f, D, k)
and the c({vert}D{vert}){sup}2. Here, the value of the constant is given
explicitly and some interesting corollaries of the theorem are given.
Reviewed by Marie-France Vigneras