The representations of p-adic fields associated to elliptic curves

Abstract

The goal of this dissertation is to find the irreducible, admissible representation of GL(2; F) attached to an elliptic curve E over a p-adic field F. Associated to E is a 2-dimensional complex representation of the Weil-Deligne group via the action of the absolute Galois group on the Tate module. On the other hand, the local Langlands correspondence states that the 2-dimensional representations of the Weil-Deligne group are in bijection with equivalence classes of irreducible, admissible representations of GL(2; F). We consider a Weierstrass equation for E of the form y2 + a1xy + a2y = x3 + a2x2 + a4x + a6 (W) We will determine the representation the p-adic field in terms of the coefficients a1; a2; a3; a4; a6. We also investigate a particular class of these reperesentations called "triply imprimitive" representations.