$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

Abstract

We discuss the ℓ-adic case of Mazur's "Program B" over Q, the problem of classifying the possible images H ≤ GL 2 (Z ℓ ) of Galois representations attached to elliptic curves E over Q, equivalently, classifying the rational points on the corresponding modular curves X H . The primes ℓ = 2 and ℓ ≥ 13 are addressed by prior work, so we focus on the remaining primes ℓ = 3, 5, 7, 11. For each of these ℓ, we compute the directed graph of arithmetically maximal ℓ-power level modular curves, compute explicit equations for most of them, and classify the rational points on all of them except X + ns (N ), for N = 27, 25, 49, 121, and two level 49 curves of genus 9 whose Jacobians have analytic rank 9.Aside from the subgroups of GL 2 (Z ℓ ) known to arise for infinitely many Q-isomorphism classes of elliptic curves E/Q, we find only 22 exceptional subgroups that arise for any prime ℓ and any E/Q without complex multiplication; these exceptional subgroups are realized by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional subgroups is complete and show that any counterexamples must arise from unexpected rational points on X + ns (ℓ) with ℓ ≥ 17, or one of the six modular curves noted above. This gives us an efficient algorithm to compute the ℓ-adic image of Galois for any non-CM elliptic curve over Q.In an appendix with John Voight we generalize Ribet's observation that simple abelian varieties attached to newforms on Γ 1 (N ) are of GL 2 -type; this extends Kolyvagin's theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of X H .

How big is the image of the Galois representations attached to CM elliptic curves?

“…In fact, it has even been conjectured that there should exist such an upper bound which does not depend on 𝐸, but only on the eld of de nition 𝐹 . This conjecture is explicitly mentioned for 𝐹 = Q in the introduction to the recent work of Rouse, Sutherland and Zureick-Brown [17], and is known to hold true under the assumption of Serre's uniformity conjecture, by previous work of Zywina (see [26,Theorem 1.4]).…”

“…its maximal abelian quotient). In order to compute the right hand side of (17), note that, if 𝐺 := Gal( 𝐿/𝐾) denotes the Galois group of the Galois closure 𝐿 of the extension 𝐾 ⊆ 𝐿, and 𝐻 𝐺 ⊆ 𝐺 denotes the normal closure of the subgroup 𝐻 := Gal( 𝐿/𝐿) inside 𝐺, then we have Gal(𝐿 /𝐾) 𝐺/𝐻 𝐺 . Since both 𝐺 and 𝐻 can be computed as subgroups of the symmetric group 𝔖 𝑛 on 𝑛 = [𝐿 : 𝐾] letters (see [7, § 6.3]), the abelian group (𝐺/𝐻 𝐺 ) ab can also be explicitly computed, for instance using the functions N C and M A in GAP [10];…”

Root Repulsion and Faster Solving for Very Sparse Polynomials Over $p$-adic Fields

“…Solving polynomial equations over the p-adic rational numbers Q p underlies many important computational questions in number theory (see, e.g., [23,8,21,47]) and is close to applications in coding theory (see, e.g., [10]). Furthermore, the complexity of solving structured equations -such as those with a fixed number of monomial terms or invariance with respect to a group action -arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [35]).…”

Congruences of Elliptic Curves Arising from Non-Surjective Mod $N$ Galois Representations

“…Therefore classifying nontrivial quadratic twist-type p l -congruences is equivalent to finding all exceptional (i.e., non-cuspidal, non-CM) rational points on the modular curves X(s p l + ) and X(ns p l + ) which do not lift to points on X(s p l ) respectively X(ns p l ). The curves X(s p l + ) have no exceptional points for p l > 7 (see [BPR13, Theorem 1.1] for p > 7 except p = 13, [BDM + 19] in the level 13 case, and [RSZ21a] when p l = 9, 25, 49). It is a well known open problem to determine the rational points on X(ns p l + ) when p l ≥ 19 (when p l = 13 and 17 this has been resolved in [BDM + 19] and [BDM + 21] respectively).…”

“…In the spirit of Mazur's "Program B" (and other work on modular curves -see e.g., [RZ15a], [SZ17], and [RSZ21a]) it would be interesting to extend Theorem 1.2 to classify all nontrivial quadratic twist-type (N, r)-congruences (assuming the conjecture that the modular curves X(ns p l + ) have no exceptional points for p l ≥ 19, see e.g., [RSZ21a, Conjecture 1.5]).…”