Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal represent...

Let F be a number field, and (ρ, V ) a continuous, n-dimensional representation of the absolute Galois group Gal(F/F ) on a finite-dimensional C-vector space V . Denote by L(s, ρ) the associated L-function, which is known to be meromorphic with a functional equation. Artin’s conjecture predicts that L(s, ρ) is holomorphic everywhere except possibly at s = 1, where its order of pole is the multi...

We study weighted pseudo almost automorphic solutions for the nonlinear fractional difference equation ∆u(n) = Au(n+ 1) + f(n, u(n)), n ∈ Z, for 0 < α ≤ 1, whereA is the generator of an α-resolvent sequence {Sα(n)}n∈N0 in B(X). We prove the existence and uniqueness of a weighted pseudo almost automorphic solution assuming that f(·, ·) is weighted almost automorphic in the first variable and sat...

We prove potential automorphy results for a single Galois representation $$G_F\rightarrow GL_n(\overline{\mathbb {Q}}_l)$$ where F is CM number field. The strategy to use the p, q switch trick go between p-adic and q-adic realisation of certain variant Dwork motive. choose this break self-duality shape motives, but not Hodge-Tate weights. Another key result that representations we come from mot...

This paper is a survey of recent work about the action of braids on selfdistributive systems. We show how the braid word reversing technique allows one to use new self-distributive systems, leading in particular to a natural linear ordering of the braids. AMS Subject Classification: 20F36, 20N02. It has been observed for many years that there exist a connection between braids and left selfdistr...

We study Whittaker functions for the principal series representation of SL(n,R). We derive a system of partial differential equations characterizing our Whittaker functions. We give explicitly power series solutions at the regular singularity of the system, and integral representations of unique moderate growth Whittaker function. Introduction In this paper we explicitly determine the radial pa...

Drinfeld in 1974, in his seminal paper [10], revolutionized the contribution to arithmetic of the area of global function fields. He introduced a function field analog of elliptic curves over number fields. These analogs are now called Drinfeld modules. For him and for many subsequent developments in the theory of automorphic forms over function fields, their main use was in the exploration of ...

This paper studies the Fourier expansion of Hecke-Maass eigenforms for GL(2,Q) of arbitrary weight, level, and character at various cusps. It is shown that the Fourier coefficients at a cusp satisfy certain very explicit multiplicativity relations. As an application, it is proved that a local representation of GL(2,Qp) which is isomorphic to a local factor of a global cuspidal automorphic repre...

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL2 is toroidal if all its right translates integrate to zero over all nonsplit tori in GL2, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one an...

In this paper we prove a version of Deligne’s conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.