We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These satisfy multiplicativity, and are expressible as products Gauss sums. The tensor product related to the Rankin-Selberg gamma factors, by which we prove that can be written exterior square relate Jacquet-Shalika Langlands-Shahidi level zero supercuspidal representatio...

We give a precise determination of the algebraicity critical values L -functions associated to Siegel modular forms half-integral weight and arbitrary degree. generalise improve on similar results for integral-weight case by adapting Rankin-Selberg method this setting, with aid Shimura's theory recent work Bouganis. An essential ingredient is proof new analogue Garrett's conjecture Klingen Eise...

We use the uniqueness of various invariant functionals on irreducible unitary representations of PGL2(R) in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an applicati...

We prove a nonvanishing result for families of GLn × GLn Rankin–Selberg L-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on GLn. A key ingredient is the regularization of the u...

Let K/F be a quadratic extension of p-adic fields. We show that a generic irreducible representation of GL(n, K) is distinguished if and ony if its Rankin-Selberg Asai L-function has an exceptional pole at zero. We use this result to compute Asai L-functions of ordinary irreducible representations of GL(2, K). In the appendix, we describe supercuspidal dihedral representations of GL(2, K) in te...

In [15], Hoffstein and Hulse defined the shifted convolution series of two cusp forms by “shifting” the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h → ∞. Our method relies on a result of Mertens and Ono [22], who showed that these values ...

The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. purpose this paper is to prove an analogue Katz attached imaginary quadratic fields via congruences between CM forms non-CM new ingredient apply Rankin-Selberg method construct a Hida family which congruent a...