2007/10/12 by Krzysztof Klosin, Klosin, Krzysztof
Mathematics · #11F33 #11F55 #11F67 #11F80 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.0710.2549
openalex publication_date 2007/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let ρf be the l-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of ρf in the following way. Let L(f,s) denote the symmetric square L-function of f. We prove that (under certain conditions) the l-adic valuation of the algebraic part of L(f, k) is no greater than the l-adic valuation of the order of S, where S is (the Pontryagin dual of) the Selmer group attached to the Galois module \ad0ρf|GK (-1), and K= Q(i). 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 U(2,2).