no. 3
On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms
[Conjecture de type BSD précisée pour des formes modulaires]
Ota, Kazuto1
1 Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043 (Japan)
Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1319-1364.

Sous certaines hypothèses, on prouve la partie rang de la conjecture précisée de Mazur–Tate de type BSD. Plus concrètement, on prouve que le rang du groupe de Selmer d’une forme modulaire elliptique est inférieur ou égal à l’ordre des zéros des éléments de Mazur–Tate, qui sont des éléments de certains algèbres de groupes construits à partir de valeurs spéciales de la fonction L associée. Notre résultat principal est considéré comme une généralisation de nos travaux antérieurs sur les courbes elliptiques.

Under some assumptions, we prove the rank-part of the Mazur–Tate refined conjecture of BSD type. More concretely, we prove that the rank of the Selmer group of an elliptic modular form is less than or equal to the order of zeros of Mazur–Tate elements, or modular elements, which are elements in certain group rings constructed from special values of the associated L-function. Our main result is regarded as a generalization of our previous work on elliptic curves.

Reçu le :
Révisé le :
Accepté le :
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DOI : 10.5802/aif.3557
Classification : 11G40, 11F11, 11R23
Keywords: Mazur–Tate refined conjectures, elliptic modular forms, $p$-adic $L$-functions
Mot clés : conjectures de Mazur-Tate précisée, formes modulaires elliptiques, fonctions $L$ $p$-adiques

Ota, Kazuto 1

1 Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Ota, Kazuto. On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms. Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1319-1364. doi : 10.5802/aif.3557. https://aif.centre-mersenne.org/articles/10.5802/aif.3557/

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