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, Tome 75 (2025) no. 1, pp. 291-329.

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
Publié le : 2025-02-10

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.
Mots-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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {An anticyclotomic {Euler} system for adjoint modular {Galois} representations},
     journal = {Annales de l'Institut Fourier},
     pages = {291--329},
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Alonso, Raúl; Castella, Francesc; Rivero, Óscar. An anticyclotomic Euler system for adjoint modular Galois representations. Annales de l'Institut Fourier, Tome 75 (2025) no. 1, pp. 291-329. doi : 10.5802/aif.3646. https://aif.centre-mersenne.org/articles/10.5802/aif.3646/

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Alonso, Raúl; Castella, Francesc; Rivero, Óscar An anticyclotomic Euler system of Hirzebruch–Zagier cycles I: Norm relations and $p$ -adic interpolation, arXiv (2025) | DOI:10.48550/arxiv.2501.15336 | arXiv:2501.15336
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