Fourier coefficients of automorphic forms, character variety orbits, and small representations

Fourier coefficients of automorphic forms, character variety orbits, and small representations

Article history: Received 8 April 2012 Revised 1 May 2012 Accepted 2 May 2012 Available online 16 August 2012 Communicated by David Goss

ar X iv : 1 20 2 . 02 10 v 1 [ m at h . N T ] 1 F eb 2 01 2 Fourier coefficients of automorphic forms , character variety orbits , and small representations

We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G2, we perform an analysis of the expansions on split real forms of E6 and E7 where simplifications take place for automorphic realizations of real representations which have small Gelfand-...

FOURIER COEFFICIENTS OF GLpNq AUTOMORPHIC FORMS IN ARITHMETIC PROGRESSIONS

Abstract. We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GLpNq Maass cusp forms for all N > 2, satisfy a central limit theorem in a suitable range, generalizing the case N “ 2 treated by É. Fouvry, S. Ganguly, E. Kowalski and P. Michel in [4]. Such universal Gaussian behaviour relies on a deep e...

On the Fourier Coefficients of Automorphic Forms on Gl2

Given E/F a quadratic extension of number fields and a cuspidal representation π of GL2(AE), we give a full description of the fibers of the Asai transfer of π. We then determine the extent to which the Hecke eigenvalues of all the Hecke operators indexed by integral ideals in F determine the representation π.

Fourier Coefficients of Gl(n) Automorphic Forms in Arithmetic Progressions