The Riemann Hypothesis for period polynomials of Hilbert modular forms

TL;DR

This paper extends the study of zeros of period polynomials satisfying a Riemann Hypothesis to Hilbert modular forms, revealing a general phenomenon across different settings.

Contribution

It introduces a natural analogue of period polynomials for Hilbert modular forms and proves the Riemann Hypothesis for their zeros.

Findings

Zeros of Hilbert modular form period polynomials satisfy a Riemann Hypothesis

The phenomenon observed in classical and cohomological cases extends to Hilbert modular forms

Supports the idea of a universal underlying principle for zeros of period polynomials

Abstract

There have been a number of recent works on the theory of period polynomials and their zeros. In particular, zeros of period polynomials have been shown to satisfy a "Riemann Hypothesis" in both classical settings and for cohomological versions extending the classical setting to the case of higher derivatives of $L$-functions. There thus appears to be a general phenomenon behind these phenomena. In this paper, we explore further generalizations by defining a natural analogue for Hilbert modular forms. We then prove that similar Riemann Hypotheses hold in this situation as well.