An anticyclotomic Euler system for adjoint modular Galois representations

Alonso, Raúl; Castella, Francesc; Rivero, Óscar

doi:10.5802/aif.3646

An anticyclotomic Euler system for adjoint modular Galois representations
[Un système d’Euler anticyclotomique pour les représentations de Galois modulaires adjointes]

Alonso, Raúl ¹ ; Castella, Francesc ² ; Rivero, Óscar ³

¹ Department of Mathematics, Princeton University, Princeton, NJ 08544-1000 (USA)
² Department of Mathematics, University of California, Santa Barbara, CA 93106 (USA)
³ Simons Laufer Mathematical Sciences Institute, 17 Gauss Way, Berkeley, CA 94720 (USA)

Annales de l'Institut Fourier, Online first, 39 p.

Résumé
Abstract

Soit $K$ un corps quadratique imaginaire et $p$ un nombre premier décomposé dans $K$ . Dans cet article, on construit un système d’Euler anticyclotomique pour le changement de base à $K$ de la représentation adjointe associée aux formes modulaires elliptiques. On relie ce système d’Euler à une fonction $L$ p-adique obtenue à partir de la construction par Eischen–Wan et Eischen–Harris–Li–Skinner des fonctions $L$ p-adiques pour les groupes unitaires. Ceci nous permet de déduire des nouveaux cas de la conjecture de Bloch–Kato en rang zéro, ainsi qu’une divisibilité vers une conjecture principale d’Iwasawa.

Let $K$ be an imaginary quadratic field and $p$ a prime split in $K$ . In this paper we construct an anticyclotomic Euler system for the adjoint representation attached to elliptic modular forms base changed to $K$ . We also relate our Euler system to a $p$ -adic $L$ -function deduced from the construction by Eischen–Wan and Eischen–Harris–Li–Skinner of $p$ -adic $L$ -functions for unitary groups. This allows us to derive new cases of the Bloch–Kato conjecture in rank zero, and a divisibility towards an Iwasawa main conjecture.

Reçu le : 2022-09-21
Révisé le : 2023-03-15
Accepté le : 2023-05-15
Première publication : 2024-07-03

DOI : 10.5802/aif.3646

Classification : 11R23, 11F85, 14G35
Keywords: Symmetric square, Euler systems, diagonal cycles, $p$-adic families of modular forms, Bloch–Kato conjecture, Iwasawa theory.
Mot clés : Carré symétrique, systèmes d’Euler, cycles diagonaux, familles $p$-adiques de formes modulaires, conjecture de Bloch–Kato, théorie d’Iwasawa.

Affiliations des auteurs :

Alonso, Raúl ¹ ; Castella, Francesc ² ; Rivero, Óscar ³

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     title = {An anticyclotomic {Euler} system for adjoint modular {Galois} representations},
     journal = {Annales de l'Institut Fourier},
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     year = {2024},
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     language = {en},
     note = {Online first},
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Alonso, Raúl; Castella, Francesc; Rivero, Óscar. An anticyclotomic Euler system for adjoint modular Galois representations. Annales de l'Institut Fourier, Online first, 39 p.

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