Degree bounds for projective division fields associated to elliptic modules with a trivial endomorphism ring
Alina Carmen Cojocaru1, 2 ; Nathan Jones3
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, Tome 33 (2021) no. 1, pp. 95-106.

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|.

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.

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DOI : 10.5802/jtnb.1153
Classification : 11G05, 11G09, 11F80
Mots-clés : 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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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, Tome 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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  • Alina Carmen Cojocaru; Sumita Garai Obstructions to reciprocity laws related to division fields of Drinfeld modules, Proceedings of the American Mathematical Society, Volume 151 (2023) no. 4, pp. 1379-1393 | DOI:10.1090/proc/16157 | Zbl:1521.11040
  • Alina Carmen Cojocaru; Mihran Papikian The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module, Journal of Number Theory, Volume 237 (2022), pp. 15-39 | DOI:10.1016/j.jnt.2021.03.026 | Zbl:1507.11049
  • Alina Carmen Cojocaru; Matthew Fitzpatrick The absolute discriminant of the endomorphism ring of most reductions of a non-CM elliptic curve is close to maximal, Arithmetic, geometry, cryptography and coding theory, AGC2T, 17th international conference, Centre International de Rencontres Mathématiques, Marseilles, France, June 10–14, 2019, Providence, RI: American Mathematical Society (AMS), 2021, pp. 51-57 | DOI:10.1090/conm/770/15430 | Zbl:1497.11146

Cité par 3 documents. Sources : zbMATH