Mod 4 galois representations from elliptic curves, a Brauer-Severi variety and a certain Brauer type embedding problem

Kevin M Mugo, Purdue University

Abstract

We determine the lattice of subfields contained in the fixed field [special characters omitted], where [special characters omitted] is a continuous, surjective mod 4 Galois representation. Let ρE,4 be a surjective Galois representation, induced by the action of G K on the 4-torsion points of an elliptic curve E, with invariant j0. We show that the unique S4 field extension contained in K( E[4]) is the splitting field of the principal quartic [special characters omitted]. We proceed to study principal quartic extensions L/K, and show that L/K is principal precisely when a certain Brauer-Severi variety has a K-rational point. When L/K is principal, and M/K is its normal closure, we show that the solvability of the embedding problem [special characters omitted] is completely determined by the discriminant dL/K. Moreover, we will show that if L/K is principal and the Hilbert symbol (–2,–dL/K) is trivial, then the embedding problem [special characters omitted] is solvable. Finally we will consider a theorem by Holden, in which he shows the existence of of mod 4 representations that do not arise from the 4-torsion of any elliptic curve E/B. We will show that the result is incorrect as stated and provide counterexamples.

Degree

Ph.D.

Advisors

Goins, Purdue University.

Subject Area

Mathematics

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