Eichler cohomology and zeros of polynomials associated to derivatives of\n L-functions

2017/04/09 by Nikolaos Diamantis, Larry Rolen, Diamantis, Nikolaos +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1704.02667

openalex publication_date 2017/04/09 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

In recent years, a number of papers have been devoted to the study of roots\nof period polynomials of modular forms. Here, we study cohomological analogues\nof the Eichler-Shimura period polynomials corresponding to higher\nL-derivatives. We state general conjectures about the locations of the roots\nof the full and odd parts of the polynomials, in analogy with the existing\nliterature on period polynomials, and we also give numerical evidence that\nsimilar results hold for our higher derivative "period polynomials" in the case\nof cusp forms. We prove a special case of this conjecture in the case of\nEisenstein series.\n

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• Period polynomials, derivatives of L-functions, and zeros of polynomials
• The Riemann Hypothesis for period polynomials of Hilbert modular forms
• Period-like polynomials for L-series associated with half-integral weight cusp forms
• The Riemann Hypothesis for period polynomials of cusp forms
• On a secant Dirichlet series and Eichler integrals of Eisenstein series