We construct the compatible system of l-adic representations associated to a regular algebraic cuspidal automorphic representation of GL n over a CM (or totally real) field and check local-global compatibility for the l-adic representation away from l and a finite number of rational primes above which the CM field or the automorphic representation ramifies. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.
We prove automorphy lifting theorems for residually reducible Galois representations in the setting of unitary groups over CM fields. Our methods are inspired by those of Skinner-Wiles in the setting of GL2. Contents 1 Preliminaries in commutative algebra 5 2 Automorphic forms on GL n (A F) 8 * This research was partially conducted during the period the author served as a Clay Research Fellow.
Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over ‫ޑ‬ as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves. In this paper we construct for each simply laced Dynkin diagram a coregular representation (G, V) and a family of algebraic curves over the geometric quotient V//G. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of G. In the case of type A 2 , we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over ‫.ޑ‬ 1. Introduction 2331 2. Preliminaries: Vinberg theory, stable involutions, subregular elements 2337 3. Subregular curves 2345 4. Jacobians and stabilizers of regular elements 2354 Acknowledgements 2366 References 2366 This research was partially conducted during the period the author served as a Clay Research Fellow. MSC2010: primary 20G30; secondary 11E72.
We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called 'Taylor-Wiles hypothesis'. We apply this to the problem of the modularity of elliptic curves over totally real fields.
As the simplest case of Langlands functoriality, one expects the existence of the symmetric power S n (π), where π is an automorphic representation of GL(2, A) and A denotes the adeles of a number field F . This should be an automorphic representation of GL(N, A) (N = n + 1). This is known for n = 2, 3 and 4. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for 'Schur representations', combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume F totally real, and the initial representation π of classical type.
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