An anticyclotomic Euler system for adjoint modular Galois representations
[Un système d’Euler anticyclotomique pour les représentations de Galois modulaires adjointes]
Annales de l'Institut Fourier, Tome 75 (2025) no. 1, pp. 291-329.

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.

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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.

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)
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},
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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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