Surjectivity of mod 2^n representations of elliptic curves

TL;DR

This paper investigates the conditions under which the Galois representations associated with elliptic curves over Q are surjective modulo 2^n and 3^n, extending known classifications to these cases.

Contribution

It classifies elliptic curves over Q with surjective mod 2^n and 3^n Galois representations that are not surjective mod 2^{n+1} and 3^{n+1}, completing the understanding of these exceptional cases.

Findings

Classified elliptic curves with surjective mod 2^n but not mod 2^{n+1} representations.

Extended Elkies' classification to include cases for powers of 2 and 3.

Provided criteria for surjectivity failure at these prime powers.

Abstract

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod l^n representation does not imply that for l^(n+1). Elliptic curves with surjective mod 3 but not mod 9 representation have been classified by Elkies. The purpose of this note is to do this in the other two cases.