Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture

Klosin, Krzysztof

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 81-166.

Soit ${\displaystyle k}$ un entier strictement positif divisible par 4, soit ${\displaystyle p>k}$ un nombre premier, et soit ${\displaystyle f}$ une forme modulaire elliptique de poids ${\displaystyle k-1}$, de niveau 4, et de caractère non trivial qui est propre pour les opérateurs de Hecke et ordinaire en ${\displaystyle p}$. Dans cet article, on donne des résultats sur la conjecture de Bloch-Kato pour les motifs ${\displaystyle {\text{ad}}^{0}M(-1)}$ et ${\displaystyle {\text{ad}}^{0}M\left(2\right)}$, où ${\displaystyle M}$ est le motif associé à ${\displaystyle f}$. Plus précisément, on démontre que, sous certaines hypothèses, la valuation $p$-adique de la part algébrique de la valeur de la fonction ${\displaystyle L}$ du carré symétrique de ${\displaystyle f}$ en ${\displaystyle k}$ est bornée par la valuation $p$-adique de la valeur de l’ordre du groupe de Selmer de l’adjoint de la représentation $p$-adique galoisienne associée à $f$ restreinte au corps gaussien et tordue par l’inverse du caractère cyclotomique. Notre méthode suit une idée de Ribet, dans le sens où nous introduisons une étape intermédiaire et produisons des congruences entre formes CAP et formes non CAP sur le groupe unitaire ${\displaystyle \text{U}(2,2)}$.

Let $k$ be a positive integer divisible by 4, $p>k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$) of weight $k-1$, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ${\text{ad}}^{0}M(-1)$ and ${\text{ad}}^{0}M\left(2\right)$, where $M$ is the motif attached to $f$. More precisely, we prove that under certain conditions the $p$-adic valuation of the algebraic part of the symmetric square $L$-function of $f$ evaluated at $k$ provides a lower bound for the $p$-adic valuation of the order of the Pontryagin dual of the Selmer group for the adjoint of the $p$-adic Galois representation attached to $f$ restricted to the Gaussian field and twisted by the inverse of the cyclotomic character. Our method uses an idea of Ribet, in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group $\text{U}(2,2)$.

Classification : 11F33, 11F55, 11F67, 11F80
Keywords: Automorphic forms on unitary groups, congruences, Selmer groups, Bloch-Kato conjecture
Mot clés : formes automorphes sur groupes unitaires, congruences, groups de Selmer, conjecture de Bloch-Kato

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