On $p$-adic analogues of the Birch and Swinnerton-Dyer conjecture for Garrett $L$-functions

Bertolini, Massimo; Seveso, Marco Adamo; Venerucci, Rodolfo

doi:10.5802/aif.3726

p

-adic analogues of the Birch and Swinnerton-Dyer conjecture for Garrett

L

-functions
[Sur les analogues

p

-adiques de la conjecture de Birch et Swinnerton-Dyer pour les fonctions

L

de Garrett]

Bertolini, Massimo ¹ ; Seveso, Marco Adamo ² ; Venerucci, Rodolfo ²

¹ Universität Duisburg-Essen, Fakultät für Mathematik, Mathematikcarrée Thea-Leymann-Straße 9, 45127 Essen (Germany)
² Università degli Studi di Milano, Dipartimento di matematica Federigo Enriques, Via Cesare Saldini 50, 20133 Milano (Italy)

Annales de l'Institut Fourier, Online first, 37 p.

Abstract (VO)
Résumé

This article formulates a $p$ -adic analogue of the Birch and Swinnerton-Dyer conjecture for a $p$ -adic $L$ -function associated to a triple of Hida families of modular forms. This involves the construction of a $p$ -adic regulator, obtained by building on Nekovář theory of Selmer complexes. Moreover, it is proved that our conjectures imply the “Elliptic Stark Conjectures” of Darmon, Lauder and Rotger.

Cet article formule un analogue $p$ -adique de la conjecture de Birch et Swinnerton-Dyer pour une fonction $L$ $p$ -adique associée à un triplet de familles de Hida de formes modulaires. Cela nécessite la construction d’un régulateur $p$ -adique, obtenu en s’appuyant sur la théorie des complexes de Selmer de Nekovář. De plus, il est prouvé que nos conjectures impliquent les « Elliptic Stark Conjectures » de Darmon, Lauder et Rotger.

Reçu le : 2022-07-03
Révisé le : 2023-03-27
Accepté le : 2023-12-21
Première publication : 2025-08-01

DOI : 10.5802/aif.3726

Classification : 11G05, 11G40
Keywords: $p$-adic BSD conjectures, $p$-adic regulators, triple product $L$-functions, rational points on elliptic curves
Mots-clés : Conjectures $p$-adiques de BSD, régulateurs $p$-adiques, fonctions $L$ triple produit, points rationnels sur les courbes elliptiques

Affiliations des auteurs :

Bertolini, Massimo ¹ ; Seveso, Marco Adamo ² ; Venerucci, Rodolfo ²

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     title = {On $p$-adic analogues of the {Birch} and {Swinnerton-Dyer} conjecture for {Garrett} $L$-functions},
     journal = {Annales de l'Institut Fourier},
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     note = {Online first},
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Bertolini, Massimo; Seveso, Marco Adamo; Venerucci, Rodolfo. On $p$-adic analogues of the Birch and Swinnerton-Dyer conjecture for Garrett $L$-functions. Annales de l'Institut Fourier, Online first, 37 p.

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