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The Riemann hypothesis for period polynomials of cusp forms

Volume 218 / 2025

William Craig, Wissam Raji Acta Arithmetica 218 (2025), 175-195 MSC: Primary 11F11; Secondary 11F67 DOI: 10.4064/aa240320-16-9 Published online: 8 January 2025

Abstract

We consider the period polynomials $r_f(z)$ associated with cusp forms $f$ of weight $k$ on all of $\mathrm {SL}_2(\mathbb {Z})$, which are the 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 $k \geq 120$, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.

Authors

  • William CraigMathematics Department
    United States Naval Academy
    Annapolis, MD, USA
    e-mail
  • Wissam RajiNumber Theory Research Unit
    Center for Advanced Mathematical Sciences (CAMS)
    Department of Mathematics
    American University of Beirut (AUB)
    Beirut, Lebanon
    e-mail

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