On the computation of local components of a newform

TL;DR

This paper introduces an algorithm to compute the local p-component of a newform's automorphic representation, especially in complex supercuspidal cases, enabling detailed analysis of local Galois representations.

Contribution

The paper presents a novel algorithm for explicitly computing the p-component of automorphic representations associated with newforms, including supercuspidal cases, and demonstrates how to extract local Galois data.

Findings

Algorithm successfully computes local components for various newforms.

Local components in supercuspidal cases are induced from characters of specific subgroups.

Local Galois representations can be fully determined from the computed local components.

Abstract

We present an algorithm for computing the $p$-component of the automorphic representation arising from a cuspidal newform $f$ for a prime $p$. This is equivalent to computing the restriction to the decomposition group at $p$ of the $\ell$-adic Galois representations attached to $f$ for any $\ell\neq p$. The situation is most interesting when $p^2$ divides the level of $f$, in which case the $p$-component could be supercuspidal. In the supercuspidal case, the local component is induced from an irreducible character of a compact-mod-center subgroup of $\text{GL}_2(\mathbf{Q}_p)$; our algorithm outputs both the group and the irreducible character. We provide examples which illustrate how the local Galois representation can be completely read off from the local component.