Computing images of Galois representations attached to elliptic curves

TL;DR

This paper introduces probabilistic algorithms to compute Galois images of elliptic curves' torsion points, enabling extensive analysis of non-CM elliptic curves over number fields with applications to databases and conjectural classifications.

Contribution

The paper develops two novel probabilistic algorithms for determining Galois images of elliptic curves' torsion points for all primes simultaneously, with polynomial and quasi-linear running times under GRH.

Findings

Applied algorithms to 140 million curves, identifying 63 exceptional Galois images over Q.

Conjecturally complete classification of Galois images for non-CM elliptic curves over quadratic fields.

Examples of exceptional images only arising for irrational j-invariants over quadratic fields.

Abstract

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein--Watkins database, some 140 million curves of conductor up to 10^10, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.