Period polynomials, derivatives of $L$-functions, and zeros of polynomials

TL;DR

This paper surveys the role of period polynomials in understanding $L$-function values and zeros, emphasizing Eichler cohomology and including derivatives and non-cuspidal forms.

Contribution

It introduces a unified framework for incorporating derivatives and non-cuspidal forms into the study of period polynomials via Eichler cohomology.

Findings

Analysis of zero locations of period polynomials

Inclusion of derivatives of $L$-functions in the framework

Extension to non-cuspidal modular forms

Abstract

Period polynomials have long been fruitful tools for the study of values of $L$-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of $L$-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for $L$-derivatives.