[Un système d’Euler anticyclotomique pour les représentations de Galois modulaires adjointes]
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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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

@article{AIF_2025__75_1_291_0, author = {Alonso, Ra\'ul and Castella, Francesc and Rivero, \'Oscar}, title = {An anticyclotomic {Euler} system for adjoint modular {Galois} representations}, journal = {Annales de l'Institut Fourier}, pages = {291--329}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {75}, number = {1}, year = {2025}, doi = {10.5802/aif.3646}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3646/} }
TY - JOUR AU - Alonso, Raúl AU - Castella, Francesc AU - Rivero, Óscar TI - An anticyclotomic Euler system for adjoint modular Galois representations JO - Annales de l'Institut Fourier PY - 2025 SP - 291 EP - 329 VL - 75 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3646/ DO - 10.5802/aif.3646 LA - en ID - AIF_2025__75_1_291_0 ER -
%0 Journal Article %A Alonso, Raúl %A Castella, Francesc %A Rivero, Óscar %T An anticyclotomic Euler system for adjoint modular Galois representations %J Annales de l'Institut Fourier %D 2025 %P 291-329 %V 75 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3646/ %R 10.5802/aif.3646 %G en %F AIF_2025__75_1_291_0
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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