Entanglement in the family of division fields of elliptic curves with complex multiplication

Campagna, Francesco; Pengo, Riccardo

doi:10.48550/arxiv.2006.00883

Cited by 4 publications

(4 citation statements)

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“…As a consequence, they also classify all elliptic curves E/Q and integers m, n such that the m-th and n-th division fields coincide. Recently, Campagna-Pengo [CP20] have studied the entanglements of CM elliptic curves focusing on when division fields become linearly disjoint. Finally, Jones-McMurdy [JM20] determine the genus zero modular curves and their j-maps whose rational points correspond to elliptic curves with entanglements of non-abelian type.…”

Section: Overview Of Proof Of Theorem Amentioning

confidence: 99%

A group theoretic perspective on entanglements of division fields

Daniels¹,

Morrow²

2020

Preprint

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In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer n and a subgroup G ⊆ GL 2 (Z/nZ) with surjective determinant, we provide a definition for G to represent an (a, b)-entanglement and give additional criteria for G to represent an explained or unexplained (a, b)-entanglement.Using these new definitions, we determine the tuples ((p, q), T ), with p < q ∈ Z distinct primes and T a finite group, such that there are infinitely many non-Q-isomorphic elliptic curves over Q with an unexplained (p, q)-entanglement of type T . Furthermore, for each possible combination of entanglement level (p, q) and type T , we completely classify the elliptic curves defined over Q with that combination by constructing the corresponding modular curve and j-map.

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Section: Overview Of Proof Of Theorem Amentioning

confidence: 99%

A group theoretic perspective on entanglements of division fields

Daniels¹,

Morrow²

2020

Preprint

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“…Program B has enjoyed substantial progress: for prime level see [BP11, BPR13, Zyw15a, Sut16, BDM + 19, LFL21], for prime power level see [RZB15,SZ17], for multi-prime level see [Zyw15b, DGJ19, GJLR16, Shi20, BJ16, Mor19, JM20, DLR19, DM20, DLRM21, Rak21], and for CM curves see [BC20,LR18,CP20,Lom17].…”

Section: Introductionmentioning

confidence: 99%

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

Rouse,

Sutherland,

Zureick-Brown

2021

Preprint

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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 .

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“…While the case N = ℓ k follows easily from the existence of non-trivial scalars in the image of Galois, the general case introduces a number of additional complications, connected with the possible 'entanglement' of torsion fields at different primes. Since not even the classification of possible ℓ-adic images is complete, the problem of describing all possible entanglements between torsion fields seems to be out of reach for the moment (but see [42], [13, §3], [12] and [18] for some positive results), so the computation of H 1 (G ∞ , E[N ]) cannot be approached directly. We are still able to obtain useful information on this group (in particular, prove Theorem 4.8) by using the inflation-restriction exact sequence and controlling the amount of entanglement by using our results on scalars and the uniform bound on the degrees of prime-degree isogenies (Mazur's theorem).…”

Section: Introductionmentioning

confidence: 99%