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

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 $\mathrm{GSp}_4$). As an application of our result, we prove a minimal $R=\mathbb{T}$ 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 $r_\pi$ associated to an automorphic representation $\pi$ of $G(\mathbb{A}_F)$. Our approach combines deformation theory with automorphic methods, providing new evidence towards the Langlands program for orthogonal and symplectic groups.