An explicit open image theorem for products of elliptic curves

Abstract

Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform results for the degrees of the Kummer extensions [K(E[N ], N −1 α) : K(E[N ])]. When K = Q, and under a minimal assumption on α, we show that the inequality [Q(E[N ], N −1 α) : Q(E[N ])] cN 2 holds with a constant c independent of both E and α.We shall often use divisibility conditions involving the symbols ℓ ∞ (where ℓ is a prime) and ∞. Our convention is that every power of ℓ divides ℓ ∞ , every positive integer divides ∞, and ℓ ∞ divides ∞.Recall from the Introduction that we denote by K M the field K(E[M ]) generated by the coordinates of the M -torsion points of E, and by K M,N (for N | M ) the field K(E[M ], N −1 α). We extend this notation by setting K ℓ ∞ = n K ℓ n , K ∞ = M K M , and more generally, forIf H is a subgroup of GL 2 (Z ℓ ), we denote by Z ℓ [H] the sub-Z ℓ -algebra of Mat 2 (Z ℓ ) generated by the elements of H.Let G be a (profinite) group. We write G ′ for its derived subgroup, namely, the subgroup of G (topologically) generated by commutators, and G ab = G/G ′ for its abelianisation, namely, its largest abelian (profinite) quotient. We say that a finite simple group S occurs in a profinite group G if there are closed subgroups H 1 , H 2 of G, with H 1 ⊳ H 2 , such that H 2 /H 1 is isomorphic to S. Finally, we denote by exp G the exponent of a finite group G, namely, the smallest integer e 1 such that g e = 1 for every g ∈ G.2.2. The ℓ-adic and adelic failures. We start by observing that it is enough to restrict our attention to the case N = M :Remark 2.1. Suppose that there is a constant C 1 satisfying M 2 [K M,M : K M ] divides C for all positive integers M . Then for any N | M , since [K M,M : K M,N ] divides (M/N ) 2 , we have that

Pink-type results for general subgroups of \operatorname{SL}_2(\mathbb{Z}_\ell )^n

“…While it is certainly true that both this theorem and its proof are quite technical, it should be remarked that this statement does enable us to show exactly the kind of 'large image' results we alluded to: the case n = 1 has been used in [Lom14] to show an explicit open image theorem for elliptic curves (without complex multiplication), and in [Lom15] we apply the case n = 2 to extend this result to arbitrary products of non-CM elliptic curves.…”

Quantitative upper bounds related to an isogeny criterion for elliptic curves

“…240] (see also its restatements [6, Theorem 7, p. 693, and Corollary 8, p. 694])). We will use this result to obtain upper bounds for the right-hand side of (13).…”

Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves

“…For now, we fix a prime such that The existence of infinitely many such primes is ensured by [Se72, Théorème 6, p. 324] and [Lo16, Theorem 1.1, p. 387] under our hypotheses that are without complex multiplication and pairwise non-isogenous over . Indeed, from [Lo16, Theorem 1.1, p. 387], we infer that there exists a least positive integer such that equals the inverse image under the canonical projection of . In particular, this means that for any prime .…”