The “Galois representations” of the title are modular representations ρ of the Galois groups of a number field F , and the ”automorphic realization” of the title refers to obtaining these representations as constituents of Galois representations attached to automorphic representations of general linear groups over F . The present article refines the moduli-theoretic arguments of [HST] to show t...

Let K be a field of characteristic not p (an odd prime), containing a primitive p-th root of unity ζ, and let L = K[z] with x n − a the minimal polynomial of z over K: thus L|K is a Kummer extension, with cyclic Galois group G = 〈σ〉 acting on L via σ(z) = ζz. T. Kohl, 1998, showed that L|K has pn−1 Hopf Galois structures. In this paper we describe these Hopf Galois structures.

The category of finite dimensional modules for the proalgebraic differential Galois group of the differential Galois theoretic closure of a differential field F is equivalent to the category of finite dimensional F spaces with an endomorphism extending the derivation of F . This paper presents an expository proof of this fact modeled on a similar equivalence from polynomial Galois theory, whose...

We construct and study two series of curves whose Jacobians admit complex multiplication. The curves arise as quotients of Galois coverings of the projective line with Galois group metacyclic groups Gq,3 of order 3q with q ≡ 1 mod 3 an odd prime, and Gm of order 2 . The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the C...

the direct limit taken over all finite Galois extensions K of k in which the integral closure Y of X is unramified over X, where GKJk denotes the Galois group of such an extension, and where GY denotes the group of points of G with coordinates in Y. For example, if X=k, our notation coincides with that of [10]. For any X, the group H(X,C) is the r-th cohomology group of the profinite group Gjr=...

Let K be a field, K̄ its separable closure, Gal(K) = Gal(K̄/K) the (absolute) Galois group of K. Let X be an abelian variety over K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ks). It is well-known [21] that Xn ia a free Z/nZ-module of rank 2dim(X); it is also a Galois submodule in X(K̄). We write K(Xn) for the field of d...

This article studies the Albanese kernel TF (E × E), for an elliptic curve E over a p-adic field F . The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion points are F -rational, or even that the Galois module E[p] is ...