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.

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=, A= 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 𝔞A, 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= et A= 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 𝔞A 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1153
Classification: 11G05,  11G09,  11F80
Keywords: Elliptic curves, Drinfeld modules, division fields, Galois representations
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
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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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