Supercuspidal ramifications and traces of adjoint lifts at good primes

TL;DR

This paper investigates the local Brauer classes of motives attached to primitive Hecke eigenforms at supercuspidal primes, providing ramification formulas, alternative proofs, and numerical examples to deepen understanding of their arithmetic properties.

Contribution

It offers explicit ramification formulas for supercuspidal primes, an alternative proof for odd primes, and a comprehensive analysis of ramifications at prime 2, advancing the understanding of local endomorphism algebras.

Findings

Explicit ramification formulas for supercuspidal primes

Alternative proof for odd primes' ramification behavior

Numerical examples supporting theoretical results

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 all supercuspidal primes in terms of traces of adjoint lifts at auxiliary primes. We give an alternative proof of the result for odd primes $p$ obtained in [MR3391026] and write down the ramification formulas for odd unramified supercuspidal primes of level zero also removing a mild hypothesis of [MR3391026]. We also give a complete description of ramifications for $p=2$. The philosophy of adjoint lifts help us to determine the local Brauer classes at non dihedral primes by using results similar to [MR2770587]. We provide some numerical examples using {Sage} and {LMFDB} supporting some of our theorems.