On the surjectivity of Galois representations associated to elliptic curves over number fields

Abstract

Given an elliptic curve E over a number field K, the ℓ‐torsion points E[ℓ] of E define a Galois representation Gal(K¯/K)→GL2(픽ℓ). A famous theorem of Serre (Invent. Math. 15 (1972) 259–331) states that as long as E has no complex multiplication (CM), the map Gal(K¯/K)→GL2(픽ℓ) is surjective for all but finitely many ℓ. We say that a prime number ℓ is exceptional (relative to the pair (E, K)) if this map is not surjective. Here, we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of E. We show in particular that conditionally on the generalized Riemann hypothesis, the largest exceptional prime of an elliptic curve E without CM is no larger than a constant (depending on K) times log NE, where NE is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre (Inst. Hautes Études Sci. Publ. Math. (1981) 323–401).

Computing Images of Galois Representations Attached to Elliptic Curves

“…Serre's open image theorem can be made effective, and under the generalized Riemann hypothesis (GRH) reasonably good bounds on the exceptional primes are known; quasilinear in the norm of the conductor of E, by [42]. This leaves the problem of computing G E ( ).…”

“…In principle our algorithms can be implemented so that they do not rely on this hypothesis, but the running times would increase exponentially. The GRH also gives us bounds on the largest exceptional prime ℓ that can occur for a given elliptic curve E/K; the results of Larson and Vaintrob [42] give bounds that are quasi-linear in log N E , where N E is the absolute value of the norm of the conductor of E. Together these allow us to bound the norms of the primes p that we must consider by a polynomial in log f , where f denotes the maximum of the absolute values of the norms of the coefficients appearing in an integral Weierstrass equation y 2 = f (x) for E.…”

“…We are thus led to the problem at hand: given an elliptic curve E/K without CM, determine the set S E of exceptional primes ℓ and the groups G E (ℓ) for each prime ℓ ∈ S E . Serre's open image theorem can be made effective, and under the generalized Riemann hypothesis (GRH) reasonably good bounds on the exceptional primes ℓ are known; quasi-linear in the norm of the conductor of E, by [42]. This leaves the problem of computing G E (ℓ).…”

On the effective version of Serre's open image theorem

“…However, even over Q, it is ineffective in the sense that no method is presently known for computing C 3 . In order to compute C 3 via the proof in [13], one would need to understand the rational points on the modular curve X ns (ℓ) for some prime ℓ ≥ 53.…”

Fields generated by torsion points of elliptic curves

“…Remark 4.4. As mentioned in the Introduction, if K is a number field and E has no complex multiplication, then one expects the equality to hold for almost all primes p (for a recent bound on exceptional primes for which ρ E,p is not surjective see[9]). Hence for a general number field K (which, of course, can contain ζ p or some coordinates of generators of E[p] only for finitely many p) one expects {x 1 , ζ p , y 2 } to be a minimal set of generators for K p over K (among those contained in {x 1 , x 2 , y 1 , y 2 , ζ p } ).…”