$p^r$-Selmer companion modular forms

Jha, Somnath; Majumdar, Dipramit; Shekhar, Sudhanshu

doi:10.5802/aif.3392

p^{r}

-Selmer companion modular forms
[

p^{r}

-Selmer compagnon formes modulaires]

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

Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 53-87.

Résumé
Abstract

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.

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.

Reçu le : 2019-02-21
Révisé le : 2019-08-06
Accepté le : 2019-12-20
Publié le : 2021-06-07

DOI : 10.5802/aif.3392

Classification : 11F33, 11R23, 11R34, 11S25, 11G40
Keywords: residual Bloch–Kato Selmer group, congruence of modular forms
Mot clés : Groupe résiduel Bloch–Kato Selmer, congruence des formes modulaires

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3392/}
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Jha, Somnath; Majumdar, Dipramit; Shekhar, Sudhanshu. $p^r$-Selmer companion modular forms. Annales de l'Institut Fourier, Tome 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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