Degree bounds for projective division fields associated to elliptic modules with a trivial endomorphism ring

Alina Carmen Cojocaru; Nathan Jones

doi:10.5802/jtnb.1153

Alina Carmen Cojocaru ^1,²; Nathan Jones ³

¹ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, 60607, IL, USA
² Institute of Mathematics “Simion Stoilow” of the Romanian Academy 21 Calea Grivitei St Bucharest, 010702 Sector 1, Romania
³ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 S Morgan St, 322 SEO Chicago, IL 60607, USA

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 95-106.

Abstract
Résumé

Let $k$ be a global field, let $A$ be a Dedekind domain with $Quot (A) = k$ , and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ that are defined over $K$ and that have a trivial endomorphism ring, with $k = ℚ$ , $A = ℤ$ in the former case and with $k$ a global function field, $A$ its ring of functions regular away from a fixed prime in the latter case, we prove, for any nonzero ideal $𝔞 ⊲ A$ , best possible estimates in the norm $| 𝔞 |$ for the degree over $K$ of the subfield of the $𝔞$ -division field of $M$ fixed by the scalars.

Soient $k$ un corps global, $A$ un anneau de Dedekind avec $Quot (A) = k$ et $K$ un corps de type fini. Pour les courbes elliptiques et les modules de Drinfeld $M$ définis sur $K$ et ayant un anneau d’endomorphismes trivial (où $k = ℚ$ et $A = ℤ$ dans le premier cas et $k$ est un corps de fonctions global et $A$ son anneau des fonctions régulières en dehors d’un idéal premier fixé dans le second cas), nous nous intéressons au sous-corps engendré par les points de $𝔞$ -torsion associé à un idéal non nul $𝔞 ⊲ A$ et à son sous-corps maximal fixé par les automorphismes scalaires. En utilisant une approche unifiée, nous prouvons les meilleures estimations possibles pour le degré de ce dernier corps sur $K$ en termes de la norme $| 𝔞 | .$

Received: 2020-02-24
Revised: 2020-10-17
Accepted: 2020-12-12
Published online: 2021-05-21

DOI: 10.5802/jtnb.1153

Classification: 11G05, 11G09, 11F80
Keywords: Elliptic curves, Drinfeld modules, division fields, Galois representations

Author's affiliations:

Alina Carmen Cojocaru ^{1, 2}; Nathan Jones ³

¹ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, 60607, IL, USA
² Institute of Mathematics “Simion Stoilow” of the Romanian Academy 21 Calea Grivitei St Bucharest, 010702 Sector 1, Romania
³ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 S Morgan St, 322 SEO Chicago, IL 60607, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Degree bounds for projective division fields associated to elliptic modules with a trivial endomorphism ring},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Alina Carmen Cojocaru; Nathan Jones. Degree bounds for projective division fields associated to elliptic modules with a trivial endomorphism ring. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 95-106. doi : 10.5802/jtnb.1153. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1153/

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