Central Values Of Rankin-Selberg L-Functions And Its Applications

Special values of L-functions arc expected to carry deep arithmetic or geo-metric information on relevant objects that are used to define the L-functions. Particular attention has been paid to central values of L-functions due to their ap-pearance in various contexts with vanishing or non-vanishing assumptions. One of the influential examples, both for the history of the L-functions and as a still-open research problem, is the conjecture developed by Birch and Swinnerton-Dyer in the early 1960s, which predicts that for an elliptic curve over Q the order of vanishing r of its Hasse-Weil L-function L(s, E) at the center of the critical strip is equal to the rank of the group of rational points E(Q). Another example is the surprising link discovered by Iwaniec and Sarnak between the proportion of non-vanishing of central values of various families of L-functions and the Gauss class number problem (see Iwaniec and Sarnak [15]).In this thesis, we are concerned with the question of determination of cusp forms by central values of L-functions. In 1997, Luo and Ramakrishnan [22] asked the question that to what extent modular forms are actually characterized by their special L-values. In the same paper, they showed that a cuspidal nor-malized newform f is uniquely determined by the family{L(1/2,f, Xd)} for all quadratic characters Xd.The work of Luo and Ramakrishnan resembles the con-verse theorem of modular forms (sec Weil [29] or Chapter 7 in Iwaniec [12]) which determines the modularity of a Dirichlet series in terms of functional equations. Afterwards, Chinta and Diaconu [3] further generalized Luo and Ramakrishnan's result to cusp forms on GL(3).Replacing twisting GL(1) objects Xd by twisting GL(2) holomorphic cusp forms, Luo [21] proved the following. Let f and f' be two normalized new forms of weight 2k (resp.2k') and level N (resp. N'). Suppose there exists a positive integer l and infinitely many primes p, such that for all forms h in the Hecke basis H2l (p) of new forms of weight 21 and level p, Then k=k', N=N' and f=f'. This is to determine holomorphic cusp forms by the central values of its twisted families of L-functions varying in level. Recently, Ganguly, Hoffstein and Sengupta [7] studied the case of determining holomorphic cusp forms by the central values of its twisted families of L-functions varying in weight. Precisely, let Hk(1) denote a Hecke basis of the space of holomorphic cusp forms of weight k for SL2(Z). Suppose that h?Hl(1) and h'?Hl'(1). If for all f?Hk(1) for infinitely many k with k sufficiently large, then l=l' and h=h'. For more results about the question of determination of cusp forms by central L-values, see Li [18], Liu [20], Luo and Ramakrishnan [23], Munshi [24] and Stark [27].The first object of this dissertation is to consider determining Hecke-Maass cusp forms by central L-values of its twists by families of holomorphic cusp forms varying in weight. Let u be a Hecke-Maass cusp form with Laplace eigenvalue?=1/4+t2 for SL2(Z). It is proved that u is uniquely determined by the central values of Rankin-Selberg L-functions L(s,f(?)u) as f runs over Hk(1) for infinitely many k with k sufficiently large. More precisely, we have the following theorem.Theorem 1.1 Suppose u and u' are fixed Hecke-Maass cusp forms for SL2(Z) with Laplace eigenvalues 1/4+t2 and 1/4+t'2 respectively. Assume that u and u' are normalized such that the first Fourier coefficients are 1. If for all f?Hk(1) for infinitely many k with k sufficiently large, then t=t' and u=u'.The second object of this dissertation is to consider determining holomorphic cusp forms by central L-values of its twists by families of Hecke-Maass cusp forms. Let f be a holomorphic cusp form of weight k for SL2(Z). It is proved that f is uniquely determined by the family{L(1/2,f(?)uj):uj?u}, where u={uj:j?1} is an orthonormal basis of Hecke-Maass cusp forms for SL2(Z). More precisely, we have the following theorem.Theorem 1.2 Let f and f' be two holomorphic Hecke eigen cusp forms for SL2(Z) of weight k and k', respectively. Let u={uj:j?1} be an orthonormal basis of Hecke-Maass cusp forms for SL2(Z). If for all uj?u, then f=f' and k=k'.The proof of Theorem 1.1 and Theorem 1.2 is based on strong multiplicity one theorem (see Piatetski-Shapiro [25]), which is a cornerstone of the theory of automorphic forms. To show the identity of two Hecke cusp forms, it is enough to prove that for all but finitely many primes p, the normalized Fourier coefficients at p are the same. In Chapter 1 of this thesis, we will prove that Theorem1.1 and Theorem 1.2 follow from the following Theorem 1.3 and Theorem 1.4, respectively.Theorem 1.3 Let f and u be as in Theorem 1.1. Let h be a real valued function, which is smooth, compactly supported on [1,2], and satisfies h(j)<<j1. Then for K sufficiently large, we have with where?f(p) is the normalized Fourier coefficient of f at prime p and h denotes the Fourier transform of h. Here?0 is Euler's constant.Theorem 1.4 Let f and U be as in Theorem 1.2. Then for T sufficiently large, we have where?j(p) is the normalized Fourier coefficient of uj at prime p.We will prove Theorem 1.3 in Chapter 3. The main idea is, as expected, to express central values of Rankin-Selberg L-functions in terms of a rapidly decay-ing series built from Fourier coefficients as in Ganguly, Hoffstein and Sengupta [7]. However, due to the presence of the extra summation over weight k, we need to use different methods from [7]. The proof of Theorem 1.3 starts from the approximate functional equation and Petersson trace formula. Subsequently, the so-called diagonal term from Petersson trace formula gives the main term and the error term in Theorem 1.3. For the so-called non-diagonal term, by dealing with the k-averaging of J-Bessel functions, we can show that it is negligible.We will establish Theorem 1.4 in Chapter 4. The main idea is the same as that of Theorem 1.3. However, for families of Maass cusp forms, the proof is different from the holomorphic case and more subtle issues arise. To prove Theorem 1.4, we apply the approximate functional equation of Rankin-Selberg L-function firstly, and then use Kuznetsov trace formula. Then we need to bound the continuous spectrum, compute the so-called diagonal term and estimate the so-called non-diagonal term from Kuznetsov trace formula. We will compute the diagonal term by contour integration and the residue theorem. For the non-diagonal term, we use some ideas of Li [19] in her recent study of central values of GL(3)×GL(2) L-functions.