An infinite family of Serre curves

Daniels, Harris B.

doi:10.1016/j.jnt.2015.03.016

Cited by 12 publications

(12 citation statements)

References 10 publications

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“…We would like to construct explicit families of elliptic curves with non-monogenic torsion fields. Daniels [9] has shown the family…”

Section: Explicit Familiesmentioning

confidence: 99%

Non-monogenic Division Fields of Elliptic Curves

Smith¹

2020

Preprint

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For various positive integers n, we show the existence of infinite families of elliptic curves over Q with n-division fields, Q(E[n]), that are not monogenic, i.e., the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every E/Q without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with a simple algorithm based on ideas of Dedekind.

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“…We would like to construct explicit families of elliptic curves with non-monogenic torsion fields. Daniels [9] has shown the family…”

Section: Explicit Familiesmentioning

confidence: 99%

Non-monogenic Division Fields of Elliptic Curves

Smith¹

2020

Preprint

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“…G7 13 ( 3 0 12 9 ) , ( 2 0 0 2 ) , ( 9 5 0 6 ) (−1/9, 1/2) −2 15 Table 15. Examples out of LMFDB: Genus X G > 1…”

Section: Appendix: Tablesmentioning

confidence: 99%

“…where ∆ E is the discriminant of the minimal model of E, and [GL 2 (Z/mZ) : G E (m)] = 2 (cf. [15]).…”

Section: Introductionmentioning

confidence: 99%

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Serre’s Constant of Elliptic Curves Over the Rationals

Daniels

González–Jiménez

2019

Experimental Mathematics

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Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre's constant associated to E, that gives necessary conditions to conclude that ρE,m, the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying valp(m) ≥ valp(A(E)) > 0 then ρE,m is non-surjective. Conditionally under Serre's Uniformity Conjecture, we determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over Q, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of GL2(Zp). Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constants.Serre's Uniformity Question. If E/Q is a non-CM elliptic curve, then must it be that ρ E,p is surjective for any prime p ≥ 41?Nowadays, an affirmative answer to the above question has received the name of Serre's Uniformity Conjecture (or sometimes just Uniformity Conjecture) despite the fact that Serre himself never conjectured it to be true.

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“…If f is a weight 2 eigenform with trivial nebentype whose coefficients are all rational, then by the Eichler-Shimura construction, there is an elliptic curve E/Q associated to f . In [3], Daniels constructed an explicit infinite family of elliptic curves over Q whose adelic Galois representations have maximal image; in particular, they have no exceptional primes. In fact, Duke and Jones have shown that, in an appropriate sense, almost all elliptic curves have no exceptional primes [4,7].…”

Section: Introductionmentioning

confidence: 99%