Representations attached to elliptic curves with a non-trivial odd torsion point

TL;DR

This paper classifies automorphic representations associated with rational elliptic curves that have a non-trivial odd torsion point, providing explicit conditions for reduction types based on curve parameters.

Contribution

It offers a complete classification of automorphic representations for these elliptic curves, linking torsion points, reduction types, and automorphic forms.

Findings

Criteria for split and non-split multiplicative reduction

Conditions for additive reduction based on parameters

Explicit classification of automorphic representations

Abstract

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and sufficient conditions on the parameters to determine when split or non-split multiplicative reduction occurs. Using this and the known results on when additive reduction occurs for these parametrized curves, we classify the automorphic representations in terms of the parameters.