Determining Of GL(2) Maass Cusp Forms By Central Values Of Rankin-Selberg L-functions

Determining modular forms by central values of L-function is an interest-ing problem which attracts a lot of attentions, many results was obtained in this direction (see [3] [6] [25] for various cases). Assuming the modularity of an L-function, Luo and Ramakrishnan [23] initially proved that the correspond-ing cusp form is uniquely determined by the special values of infinitely many twisted L-functions. Precisely, let f and g be two normalized cuspidal new-forms, then f=g if L(1/2, f(?)χd)= cL(1/2, g(?)χd) is true for all quadratic charactersχd.Instead of twisting by quadratic characters, Luo [22] showed the following fact. Let f and f' be two normalized newforms of weight 2k (resp.2k') and level N (resp. N'). Suppose that there exists a positive integer l and infinitively many primes p, such that for all forms g in the Hecke basis H2l(Γ0(p)) of newforms of weight 21 and level p, there holds then k=k', N=N', f=f'. This is the case that determine modular forms by the central values of its twisting families of L-functions varying in level aspect. Ganguly, Hoffstein and Sengupta [6] studied the case of determining modular forms by central values of its twisting families of L-functions varying in weight aspect. More precisely, let Hk(1) denote a Hecke basis of the space of holomorphic cusp forms of weight k for SL(2,Z). Suppose g∈Hl(1) and h∈Hv(1). If for infinitely many k, the equality holds for all f∈Hk(1),then l=l' and g=h.For GL(3)self-dual cusp forms, Chinta and Diaconu [3] showed an analogous result as in [23] by using a double Dirichlet series method.In [21],Liu proved the case of self-dual Maass-Hecke cusp forms for GL(3).In this paper,we are concerned with the case of even Maass-Hecke cusp forms for SL(2,Z).Let g be an even Maass-Hecke cusp form for SL(2,Z) associated to the Laplace eigenvalue 1/4+r2.Let U={uj:j≥1} be an orthonormal basis of Maass-Hecke forms of type sj=1/2+itj(tj≥0)in the space C(SL(2,Z)\H)of Maass cusp forms.We show that g is uniquely determined by the central values of the family {L(s,uj×g):uj∈U),where U denotes an orthonoarmal basis of Maass-Hecke forms of type sj=1/2+itj with large tj for SL(2,z).Our main result is the following theorem. Theoreml.1 Let f and 9 be even Maass-Hecke cusp forms for SL(2,Z)of type 1/2+ir and 1/2+ir',respectively.Let c be a constant.If the equality holds for all uj∈U of type 1/2+itj with tj sufficiently large,then f=g.