Supercuspidal ramifications and traces of adjoint lifts

Abstract

In this paper, we write down the local Brauer classes of the endomorphism algebras of motives attached to non-CM primitive Hecke eigenforms for the supercuspidal prime . The same for odd supercuspidal primes are determined by Bhattacharya-Ghate. We also treat the case of odd unramified supercuspidal primes of level zero also removing a mild hypothesis of them. As an intermediate step, we write down a description of the inertial Galois representation even for generalizing the construction of Ghate-M-zard. Some numerical examples using Sage and LMFDB are provided supporting some of our theorems.