We discuss the nonvanishing of the family of central values L( 1 2 , f ⊗ χ), where f is a fixed automorphic form on GL(2) and χ varies through class group characters of an imaginary quadratic field K = Q( √ −D), as D varies; we prove results of the nature that at least D1/5000 such twists are nonvanishing. We also discuss the related question of the rank of a fixed elliptic curve E/Q over the H...

Given a pair of distinct unitary cuspidal automorphic representations for GL([Formula: see text]) over number field, let [Formula: text] denote the set finite places at which are unramified and their associated Hecke eigenvalues differ. In this paper, we demonstrate how conjectures on automorphy possible cuspidality adjoint lifts Rankin–Selberg products imply lower bounds size text]. We also ob...

This is a joint work with Yangbo Ye. We prove a subconvexity bound for Rankin-Selberg L-functions L(s, f⊗g) associated with a Maass cusp form f and a fixed cusp form g in the aspect of the Laplace eigenvalue 1/4 + k2 of f , on the critical line Res = 1/2. Using this subconvexity bound, we prove the equidistribution conjecture of Rudnick and Sarnak on quantum unique ergodicity for dihedral Maass...

where the notation is as follows. Let φ(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL(2,Z), and denote by a(n) the n-th Fourier coefficient of φ(z). We suppose that φ(z) is a normalized eigenfunction for the Hecke operators T (n), that is, a(1) = 1 and T (n)φ = a(n)φ for every n ∈ N. The classical example is a(n) = τ(n) (κ = 12), the Ramanujan function defi...

Following G. Laumon [12], to a nonramified `-adic local system E of rank n on a curve X one associates a complex of `-adic sheaves nKE on the moduli stack of rank n vector bundles on X with a section, which is cuspidal and satisfies the Hecke property for E. This is a geometric counterpart of the well-known construction due to J. Shalika [19] and I. Piatetski-Shapiro [18]. We express the cohomo...

In this paper we begin the study of two Rankin-Selberg integrals defined on the exceptional group of type GE6. We show that each factorizes and that the contribution from the unramified places is, in one case, the degree 54 Euler product LS(π× τ, E6× GL2, s) and in the other case the degree 30 Euler product L S(π × τ,∧2 ×GL2, s).