p r -Selmer companion modular forms
Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 53-87.

The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves E 1 and E 2 over a number field K, Mazur–Rubin have defined them to be n-Selmer companion if for every quadratic character χ of K, the n-Selmer groups of E 1 χ and E 2 χ over K are isomorphic. Given a prime p, they have given sufficient conditions for two elliptic curves to be p r -Selmer companion in terms of mod-p r congruences between the curves. We discuss an analogue of this for Bloch–Kato p r -Selmer groups of modular forms. We compare the Bloch–Kato Selmer group of a modular form respectively with the Greenberg Selmer group when the modular form is p-ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary at p. We also indicate the relation between our results and the well-known congruence results for the special values of the corresponding L-functions due to Vatsal.

L’étude de groupes n-Selmer de courbes elliptiques sur des corps de nombres algébriques dans un passé récent a conduit à la découverte de certains résultats profonds en arithmétique des courbes elliptiques. Étant données deux courbes elliptiques E 1 et E 2 sur un corps de nombres algébriques K, Mazur–Rubin les a définies comme n-Selmer compagnon si pour chaque caractère quadratique χ de K, les groupes n-Selmer de E 1 χ et E 2 χ sur K sont isomorphes. Étant donné un nombre premier p, ils ont donné des conditions suffisantes pour que deux courbes elliptiques soient des compagnons p r -Selmer en termes de congruences mod-p r entre les courbes. Nous discutons d’un analogue de ce résultat pour les groupes p r -Bloch–Kato Selmer de formes modulaires. Nous comparons le groupe Bloch–Kato Selmer d’une forme modulaire respectivement avec le groupe Greenberg Selmer lorsque la forme modulaire est p-ordinaire et avec les groupes de Selmer signés de Lei–Loeffler–Zerbes lorsque la forme modulaire est non ordinaire en p. Nous relions aussi nos résultats de congruence bien connus pour les valeurs spéciales des fonctions L correspondantes dues à Vatsal.

Published online:
DOI: 10.5802/aif.3392
Classification: 11F33,  11R23,  11R34,  11S25,  11G40
Keywords: residual Bloch–Kato Selmer group, congruence of modular forms
Jha, Somnath 1; Majumdar, Dipramit 2; Shekhar, Sudhanshu 1

1 Department of Mathematics and Statistics IIT Kanpur Kanpur 208016, India
2 Department of Mathematics IIT Madras Chennai 600036, India
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {$p^r${-Selmer} companion modular forms},
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Jha, Somnath; Majumdar, Dipramit; Shekhar, Sudhanshu. $p^r$-Selmer companion modular forms. Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 53-87. doi : 10.5802/aif.3392. https://aif.centre-mersenne.org/articles/10.5802/aif.3392/

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