On the change of root numbers under twisting and applications

TL;DR

This paper investigates how the root number of modular forms changes under twisting, providing methods to determine local representation types at primes and discussing special cases like p=2.

Contribution

It offers a detailed analysis of root number variations under twisting and applications to classify local representations of modular forms at primes.

Findings

Root number change can be expressed via local Weil-Deligne representations.

Method to determine local type (Steinberg, Principal Series, Supercuspidal) at odd primes.

Analysis of the p=2 case where twisting alone is insufficient.

Abstract

The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime $p$, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at $p$ by analyzing the change of sign under a suitable twist. We also explain the case $p=2$, where twisting is not enough in general.