We state a qualitative form of strong multiplicity one for GL1. We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on GL(n), and for the cuspidal representations on GL(n) induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Rama...

Given a ∈ L(R) and A the generator of an L-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t) = ∫ t −∞ a(t − s)[Au(s) + f(s, u(s))]ds for each f : R × X → X almost automorphic in t, for each x ∈ X, and satisfying diverse Lipschitz type conditions. In the scalar case, we pro...

We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.

This bears upon the construction of non-trivial residual square-integrable automorphic forms coming from cuspforms on smaller groups, anticipating that such automorphic forms occur as residues of Eisenstein series. For example, we can see why there is no interesting (i.e., non-constant) non-cuspidal discrete spectrum for GL(2) nor for GL(3), but only for GL(4) and larger groups. Namely, the Eis...

In this article, first we introduce and study the concept of Sγ pseudo almost automorphy (or generalized Stepanov-like pseudo almost automorphy), which is more general than the notion of Stepanov-like pseudo almost automorphy due to Diagana. We next study the existence of solutions to some classes of nonautonomous differential equations of Sobolev type in Sγ -pseudo almost automorphic spaces. T...

(m,n)6=(0,0) y |mz+n|2s is the nonholomorphic Eisenstein series on the upper half plane, then for all y sufficiently large, E(z, s) has a ”Siegel zero.” That is E(z, β) = 0 for a real number β just to the left of one. We give a generalization of this result to Eisenstein series formed with real valued automorphic forms on a finite central covering of the adele points of a connected reductive al...

We present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor, and Berger-Harcos, we can associate to an automorphic representation a family of compatible -adic representations. Our algorithm is based on Faltings-Serre’s method to pr...

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GLn over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight. 2...