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Krzysztof Klosin
Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture
(Congruences entre formes modulaires sur U(2,2) et la conjecture de Bloch-Kato)
Annales de l'institut Fourier, 59 no. 1 (2009), p. 81-166, doi: 10.5802/aif.2427
Article PDF | Reviews MR 2514862 | Zbl 1214.11055
Class. Math.: 11F33, 11F55, 11F67, 11F80
Keywords: Automorphic forms on unitary groups, congruences, Selmer groups, Bloch-Kato conjecture

Résumé - Abstract

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 $\text{ad}^0 M(2)$, 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 $\text{U}(2,2)$.

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