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 ( ) be the image of the Galois representation induced by the action of the absolute Galois group of K on the -torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G E ( ) up to local conjugacy for all primes by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G E ( ) up to one of at most two isomorphic conjugacy classes of subgroups of GL 2 (Z/ 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, 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 10 10 , thereby obtaining a conjecturally complete list of 63 exceptional Galois images G E ( ) that arise for E/Q without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images G E ( ) 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.

Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant

“…Then E is either a base change of an elliptic curve over Q, or a quadratic twist of such a curve. By [34,Cor. 5.25], the images ρ E, (Gal(K /K )) which may arise in this case are precisely the groups G arising for base changes of elliptic curves E/Q, along with their twists: these are the group G, −1 and its index-2 subgroups that do not contain −1.…”

“…Let Excep( ) denote the set of exceptional subgroups of GL 2 ( ). The results for each subgroup in Table 1 were computed [31,34] using Magma [2], and the unions of these sets of indices over each Excep( ) are the sets I(Excep( ); M( )) appearing in Table 9. The analogous sets I(GL 2 ( ); M( )) are determined in §3.1, and appear in Table 10.…”

“…The results for each exceptional subgroup were computed [31,34] using Magma, and the div-minima of the unions of the two sets (4.1) of indices over each Excep( ) are the sets I(Excep( )) and I Q (Excep( )), respectively, listed in Table 11. The set div-min I(GL 2 ( )) is determined by Lemma 4.1 below, and appears in Table 12.…”

“…1.1] based on data from some 140 million elliptic curves, including all curves of conductor up to 400,000 [9]: Conjecture 1.1 (Sutherland) Let E/Q be an elliptic curve without complex multiplication and let be a prime. Then the image of ρ E, is either equal to GL 2 (Z/ Z) or conjugate to one of the 63 exceptional groups listed in Table 1 (see also [34,Tables 3,4]). …”

Serre’s Constant of Elliptic Curves Over the Rationals

“…Theorem 10. Assuming the Uniformity Conjecture: 3,4,5,6,7,8,9,10,11,12,13,14,15,20,21,24,28,40, 56, 104}.…”