COMPUTING IMAGES of GALOIS REPRESENTATIONS ATTACHED to ELLIPTIC CURVES

Abstract

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let GE (ℓ) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ℓ-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine GE (ℓ) up to local conjugacy for all primes ℓ by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine GE (ℓ) up to one of at most two isomorphic conjugacy classes of subgroups of GL2 (Z=ℓZ) 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 integralWeierstrass equation for E, and for our Monte Carlo algorithm, quasilinear 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 1010, thereby obtaining a conjecturally complete list of 63 exceptional Galois images GE (ℓ) that arise for E/Q without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images GE (ℓ) that 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.