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 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.

minor corrections, 47 pages

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Cited by in corpus (14)

• Computing images of Galois representations attached to elliptic curves
• Modular curves of prime-power level with infinitely many rational points
• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Complete classification of the torsion structures of rational elliptic curves over quintic number fields
• Galois representations attached to elliptic curves with complex multiplication
• Sato-Tate distributions of twists of the Fermat and the Klein quartics
• Serre's constant of elliptic curves over the rationals
• An algorithm for determining torsion growth of elliptic curves
• On the effective version of Serre's open image theorem
• Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$
• Reductions of points on algebraic groups, II
• Towards a classification of isolated $j$-invariants
• Polynomials realizing images of Galois representations of an elliptic curve
• Diophantine stability for elliptic curves on average