On Galois inertial types of elliptic curves over $\mathbb{Q}_\ell$

TL;DR

This paper explicitly classifies the inertial Weil--Deligne types associated with elliptic curves over local fields of prime characteristic, enhancing understanding of their local Galois representations.

Contribution

It provides a complete and explicit description of Galois inertial types for elliptic curves over ield of prime characteristic, filling a gap in the local Langlands correspondence.

Findings

Explicit classification of inertial types for elliptic curves over ields

Complete description of Weil--Deligne representations associated with these curves

Clarification of the local Galois representation structure

Abstract

We provide a complete, explicit description of the inertial Weil--Deligne types arising from elliptic curves over $\mathbb{Q}_\ell$ for $\ell$ prime.