Combining sieve methods with automorphic form theory and techniques from -adic cohomology, we prove that the sign of Kloosterman sums Kl(1, 1; n) changes infinitely often as n ranges over the squarefree integers having all their prime factors larger than n1/23.9 . 1. Introduction Soient a, b et n trois entiers, avec n 1. On rappelle que la somme de Kloosterman Kl(a, b; n) est d´finie par la formule e ax + bx Kl(a, b; n) = exp 2πi . n x mod n (x,n)=1 (la notation x indique l’inverse de x modulo n). Rappelons que c’est un nombre r´el, qui,...

We prove an identity of Kloosterman integrals which is the fundamental lemma of a relative trace formula for the general linear group in n variables. 1. Introduction One of the simplest examples of Langlands’ principle of functoriality is the quadratic base change. Namely, let E/F be a quadratic extension of global fields and z
→ z the corresponding Galois conjugation. The base change associates to every automorphic representation π of GL(n, F) an automorphic representation Π of GL(n,E). If n = 1 then π is an id`ele class character and Π(z) = π(zz)....