A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

TL;DR

This paper studies how the local epsilon factors of modular forms change under quadratic twisting, using representation theory to relate these changes to supercuspidal types and Morita's p-adic Gamma function.

Contribution

It introduces a novel representation-theoretic approach to analyze the variance of epsilon factors for modular forms with arbitrary nebentypus, extending previous results.

Findings

Identifies the supercuspidal representation type from epsilon factor variance.

Relates epsilon factors for ramified principal series and unramified supercuspidal representations to Morita's p-adic Gamma function.

Provides new methods differing from Pacetti's approach for trivial nebentypus forms.

Abstract

In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-adic Gamma function.