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

Campagna, Francesco; Pengo, Riccardo

doi:10.2140/pjm.2022.317.21

“…The tables that follow provide data for various subgroups H of -power level that include: the label of H as defined in Section 2.4; a list of generators for H in (which we believe is minimal); for , the label of assigned by Cummins and Pauli [CP03, CP]; the number of cusps and rational cusps , as described in Section 2; the analytic rank r of , whose computation is described in Section 6; the genus g of ; the dimensions and LMFDB labels of the Galois orbits of newforms whose modular abelian varieties are isogeny factors of (exponents denote multiplicities). …”

Section: Tablesmentioning

confidence: 99%

“…Mazur’s Program B has seen substantial progress over the past decade: for prime level, see [BP11, BPR13, Zyw15a, Sut16, BDM+19, BDM+21, LFL21]; for prime power level, see [RZB15, SZ17]; for multi-prime level, see [Zyw15b, DGJ, GJLR16, BJ16, Mor19, JM22, DLR, DM, DLRM21, Rak21, BS22]; and for CM curves, see [BC20, LR, CP, Lom17].…”

Section: Introductionmentioning

confidence: 99%

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

Rouse

¹

,

Sutherland

²

,

Zureick-Brown

³

2022

Forum of Mathematics, Sigma

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We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$ . For each of these $\ell $ , we compute the directed graph of arithmetically maximal $\ell $ -power level modular curves $X_H$ , compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$ , for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$ . Aside from the $\ell $ -adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$ -isomorphism classes of elliptic curves $E/\mathbb {Q}$ , we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$ , or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $ -adic images of Galois for any elliptic curve over $\mathbb {Q}$ . In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$ -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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“…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, and they used their results to determine the index of the adelic image of Galois associated to a CM elliptic curve over Q inside of the normalizer of a certain Cartan subgroup (see loc. cit.…”

Section: Proposition B the Genus One Modular Curves Of Positive Rankmentioning

confidence: 99%

Untitled

2022

Trans. Amer. Math. Soc. Ser. B

0

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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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“…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, and they used their results to determine the index of the adelic image of Galois associated to a CM elliptic curve over Q inside of the normalizer of a certain Cartan subgroup (see loc. cit.…”

Section: Proposition B the Genus One Modular Curves Of Positive Rankmentioning

confidence: 99%

A group theoretic perspective on entanglements of division fields

Daniels

¹

,

Morrow²

2022

Trans. Amer. Math. Soc. Ser. B

6

0

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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 n and a subgroup G ⊆ G L 2 ( Z / n Z ) G\subseteq GL_2( \mathbb {Z}/n\mathbb {Z}) with surjective determinant, we provide a definition for G G to represent an ( a , b ) (a,b) -entanglement and give additional criteria for G G to represent an explained or unexplained ( a , b ) (a,b) -entanglement. Using these new definitions, we determine the tuples ( ( p , q ) , T ) ((p,q),T) , with p > q ∈ Z p>q\in \mathbb {Z} distinct primes and T T a finite group, such that there are infinitely many non- Q ¯ \overline {\mathbb {Q}} -isomorphic elliptic curves over Q \mathbb {Q} with an unexplained ( p , q ) (p,q) -entanglement of type T T . Furthermore, for each possible combination of entanglement level ( p , q ) (p,q) and type T T , we completely classify the elliptic curves defined over Q \mathbb {Q} with that combination by constructing the corresponding modular curve and j j -map.

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

Cited by 6 publications

References 32 publications

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

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A group theoretic perspective on entanglements of division fields

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