Division in Modules and Kummer Theory
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 389-441

In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative algebraic groups. With these tools we provide an effective bound for the degree of the field extensions arising from division points of elliptic curves, extending previous results of Javan Peykar for CM curves and of Lombardo and the author for the non-CM case.

Dans ce travail, nous généralisons la notion de module injectif et développons une théorie de divisibilité pour les modules sur un anneau quelconque, ce qui fournit un cadre général et unifié pour l’étude des extensions de corps commutatifs de type Kummer provenant des groupes algébriques commutatifs. Avec ces outils, nous fournissons une borne effective pour le degré des extensions de corps commutatifs engendrées par des points de division des courbes elliptiques, en étendant les résultats précédents de Javan Peykar pour les courbes CM et de Lombardo et l’auteur pour le cas non-CM.

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DOI : 10.5802/jtnb.1326
Classification : 13C11, 16D10, 16D90, 11F80, 11G05, 11G10
Keywords: Kummer theory, elliptic curves, abelian varieties, modules, injective modules, injectivity, Galois representations, open-image theorem
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Division in {Modules} and {Kummer} {Theory}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {389--441},
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Sebastiano Tronto. Division in Modules and Kummer Theory. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 389-441. doi: 10.5802/jtnb.1326

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