Bias of Root Numbers for Hilbert Newforms of Cubic Level

TL;DR

This paper derives a general formula for the bias of root numbers in Hilbert modular newforms of cubic level, extending previous results to number fields and providing explicit calculations for specific base fields.

Contribution

It introduces a new general formula for root number bias in Hilbert modular forms over number fields, using Jacquet-Zagier's trace formula, and performs explicit calculations for certain quadratic fields.

Findings

Explicit bias formulas for specific quadratic fields

Extension of bias phenomenon to number fields

Method based on Jacquet-Zagier's trace formula

Abstract

We give a general formula of the bias of root numbers for Hilbert modular newforms of cubic level. Explicit calculation is given when the base field is $\mathbb{Q}, \mathbb{Q}(\sqrt{2}), \mathbb{Q}(\sqrt{5})$ and the level is the cube of certain rational integers. This complements a previous result of the second author and extends the bias phenomenon to the number fields. Our method is based on Jacquet-Zagier's trace formula, and the explicit calculation works generally for all real quadratic fields of narrow class number one and for rational cubic levels.