The Riemann Hypothesis for period polynomials of cusp forms

2023/05/06 by William Craig, Craig, William, Raji, Wissam
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2305.03951

Abstract

We consider the period polynomials r_f(z) associated with cusp forms f of weight k on all of SL₂( ℤ ), which are generating functions for the critical L-values of the modular L-function associated to f. In 2014, El-Guindy and Raji proved that if f is an eigenform, then r_f(z) satisfies a ``Riemann hypothesis" in the sense that all its zeros lie on the natural boundary of its functional equation. We show that this phenomenon is not restricted to eigenforms, and we provide large natural infinite families of cusp forms whose period polynomials almost always satisfy the Riemann hypothesis. For example, we show that for weights k ≥ 120, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.

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