The study of -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 and over a number field , Mazur–Rubin have defined them to be -Selmer companion if for every quadratic character of , the -Selmer groups of and over are isomorphic. Given a prime , they have given sufficient conditions for two elliptic curves to be -Selmer companion in terms of mod- congruences between the curves. We discuss an analogue of this for Bloch–Kato -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 -ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary at . We also indicate the relation between our results and the well-known congruence results for the special values of the corresponding -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 et sur un corps de nombres algébriques , Mazur–Rubin les a définies comme -Selmer compagnon si pour chaque caractère quadratique de , les groupes -Selmer de et sur sont isomorphes. Étant donné un nombre premier , ils ont donné des conditions suffisantes pour que deux courbes elliptiques soient des compagnons -Selmer en termes de congruences mod- entre les courbes. Nous discutons d’un analogue de ce résultat pour les groupes -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 -ordinaire et avec les groupes de Selmer signés de Lei–Loeffler–Zerbes lorsque la forme modulaire est non ordinaire en . Nous relions aussi nos résultats de congruence bien connus pour les valeurs spéciales des fonctions correspondantes dues à Vatsal.
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Keywords: residual Bloch–Kato Selmer group, congruence of modular forms
Mot clés : Groupe résiduel Bloch–Kato Selmer, congruence des formes modulaires
Jha, Somnath 1; Majumdar, Dipramit 2; Shekhar, Sudhanshu 1
@article{AIF_2021__71_1_53_0, author = {Jha, Somnath and Majumdar, Dipramit and Shekhar, Sudhanshu}, title = {$p^r${-Selmer} companion modular forms}, journal = {Annales de l'Institut Fourier}, pages = {53--87}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {71}, number = {1}, year = {2021}, doi = {10.5802/aif.3392}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3392/} }
TY - JOUR AU - Jha, Somnath AU - Majumdar, Dipramit AU - Shekhar, Sudhanshu TI - $p^r$-Selmer companion modular forms JO - Annales de l'Institut Fourier PY - 2021 SP - 53 EP - 87 VL - 71 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3392/ DO - 10.5802/aif.3392 LA - en ID - AIF_2021__71_1_53_0 ER -
%0 Journal Article %A Jha, Somnath %A Majumdar, Dipramit %A Shekhar, Sudhanshu %T $p^r$-Selmer companion modular forms %J Annales de l'Institut Fourier %D 2021 %P 53-87 %V 71 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3392/ %R 10.5802/aif.3392 %G en %F AIF_2021__71_1_53_0
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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