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

TL;DR

This paper provides evidence for the Bloch-Kato conjecture by establishing bounds on the l-adic valuation of special L-values through congruences between modular forms on U(2,2).

Contribution

It introduces a novel approach using congruences between CAP and non-CAP forms on U(2,2) to support the Bloch-Kato conjecture for certain Galois representations.

Findings

Bound on the l-adic valuation of algebraic L-values.

Establishment of congruences between modular forms on U(2,2).

Support for the Bloch-Kato conjecture in specific cases.

Abstract

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 \rho_f 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 \rho_f 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 \ad^0\rho_f|_{G_K} (-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).