We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations of GLn over an imaginary CM or totally real field. We prove this compatibility up to semisimplification in all cases, and up to Frobenius semisimplification i...

This paper is a continuation of [HT]. Let L be an imaginary CM field and let Π be a regular algebraic (i.e., Π∞ has the same infinitesimal character as an algebraic representation of the restriction of scalars from L to Q of GLn) cuspidal automorphic representation of GLn(AL) which is conjugate self-dual (Π ◦ c ∼= Π∨) and square integrable at some finite place. In [HT] it is explained how to at...

where A t for t ∈ R is a family of closed linear operators with domains D A t satisfying Acquistapace-Terreni conditions, and the function f : R × X → X is almost automorphic in t ∈ R uniformly in the second variable, was studied. For that, the author made extensive use of techniques utilized in 2 , exponential dichotomy tools, and the Schauder fixed point theorem. In this paper we study the ex...

We study limits of discrete series with infinitesimal character zero for Sp4: their n-cohomology and their contribution to “nonclassical” automorphic cohomology of the period domain for Hodge structures of mirror quintic type. As an application, we obtain the first generalization beyond SU(2, 1) of a result of [C1], showing that this cohomology can be reached by cup products of pairs of “classi...

Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function...

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...

1.1. Let X be a smooth, complete, geometrically connected curve over Fq. Denote by F the field of rational functions on X , by A the ring of adeles of F , and by Gal(F/F ) the Galois group of F . The present paper may be considered as a step towards understanding the geometric Langlands correspondence between n–dimensional `–adic representations of Gal(F/F ) and automorphic forms on the group G...

Serre has asked if there are motives M with motivic Galois group of type G2 [Se3; pg 386]. This paper is the first step in a project to construct such a motive M , of rank 7 and weight 0, over the base field Q. Let G be the anisotropic form of G2 over Q, and let π = ⊗̂vπv be an automorphic representation of the adelic group G(A). At almost all primes p, the local representation πp is unramified ...