Automorphic side of Taylor-Wiles method for orthogonal and symplectic groups

2024/11/07 by Zhang, Xiaoyu
#11F55 #11F70 #11F80 #11F85 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.04897

Abstract

The core of the Taylor-Wiles and Taylor-Wiles-Kisin method in proving modularity lifting theorems is the construction of Taylor-Wiles primes satisfying certain conditions relating automorphic side and Galois side. In this article, we construct such primes and develop the automorphic side of Taylor-Wiles method for definite special orthogonal or symplectic groups G over a totally real number field F, beyond the only known case for definite unitary groups (except for GSp₄). As an application of our result, we prove a minimal R=\mathbbT theorem for G, extending the scope of modularity lifting results to this setting. As a direct consequence, we deduce the Bloch--Kato conjecture for the adjoint of the Galois representation r_π associated to an automorphic representation π of G(\mathbbA_F). Our approach combines deformation theory with automorphic methods, providing new evidence towards the Langlands program for orthogonal and symplectic groups.

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