Euler Factors and Local Root Numbers for Symmetric Powers of Elliptic Curves

Abstract

For any elliptic curve E over a number field, there is, for each n � 1, a symmetric n th -power L-function, defined by an Euler product, and conjecturally having a meromorphic continuation and satisfying a precise func- tional equation. The sign in the functional equation is conjecturally a product of local signs. Given an elliptic curve over a finite extension of some Qp, we calculate the associated Euler factor and local sign, for any n � 1.