Ramanujan’s original conjecture is concerned with the estimation of Fourier coefficients of the weight 12 holomorphic cusp form ∆ for SL(2,Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigenvalues of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associa...

We consider periods of automorphic representations adele groups defined by integrals along Gelfand subgroups. define natural maps between local components such and construct corresponding global using L-functions. This leads to an introduction a invariant representation arising from two periods. compute this in some cases.

Abstract. We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation π of GSp 4 (F ), where F is a non-archimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in π to be a test vector. We apply...

We establish the existence of congruences between a fixed endoscopic, globally generic, cuspidal automorphic representation Π GSp(4) square-free conductor and stable representations same weight modulo certain prime factors value at 1 adjoint L-function Π, normalized by suitable period.

We derive a precise relation of poles Eisenstein series associated to the cuspidal datum $\chi\otimes\sigma$ and lowest occurrence theta lifts automorphic representation $\sigma$ unitary group, where $\chi$ is conjugate self-dual character. A key ingredient proof computation period integrals truncated series.

We prove the indecomposability of the Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form over a totally real field F under the assumption that either the degree of F over Q is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of F .

Let π be the automorphic representation of GSp4(A) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and τ be an arbitrary cuspidal, automorphic representation of GL2(A). Using Furusawa’s integral representation for GSp4 ×GL2 combined with a pullback formula involving the unitary group GU(3, 3), we prove that the L-functions L(s, π× τ ) are “nice”. The conve...

In 1980s, Gross–Zagier [GZ86] established a formula that relates the Neron–Tate height of Heegner points on modular curves to the central derivative of certain L-functions associated to modular forms. Around the same time, Waldspurger proved a formula, relating toric periods of modular forms to the central value of certain L-functions. Gross put both of these formula in the framework of represe...

Let $$\pi $$ be a non-self-dual unitary cuspidal automorphic representation not of solvable polyhedral type for $$\mathrm{GL}(2)$$ over number field. We show that has positive upper Dirichlet density Hecke eigenvalues in any sector whose angle is at least 2.63 radians.

Let π be a regular algebraic cuspidal automorphic representation of GL2 over an imaginary quadratic number field K, and let ` be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible `-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explici...