Riemann hypothesis for period polynomials of modular forms

• Type of Material: Article
• Authors: Seokho Jin, Korea Institute for Advanced Study; Wenjun Ma, Shandong University; Ken Ono, Emory University; Kannan Soundararajan, Stanford University
• Language: English
• Date: 2016-03-08
• Publisher: National Academy of Sciences
• Publication Version: Final Published Version
• Copyright Statement: 2016 National Academy of Sciences
• Final Published Version (URL): http://dx.doi.org/10.1073/pnas.1600569113
• Title of Journal or Parent Work: Proceedings of the National Academy of Sciences
• Volume: 113
• Issue: 10
• Start Page: 2603
• End Page: 2608
• Grant/Funding Information: K.O. acknowledges the support of the Asa Griggs Candler Fund and the National Science Foundation (NSF).; S.J. thanks the Korea Institute for Advanced Study for its generous support.; W.M. thanks the China Scholarship Council for its generous support.; K.S. acknowledges the support of the NSF and the Simons Foundation for a Simons Investigator Grant.
• Keywords: Birch and Swinnerton-Dyer conjecture; arithmetic geometry; Riemann hypothesis; period polynomialsperiod polynomials; circle of symmetry; modular L-function; Bloch–Kato conjecture
• Research Categories: Education, Mathematics

Abstract

The period polynomial rf(z) for an even weight k ≥ 4 newform f ∈ Sk(Γ0(N)) is the generating function for the critical values of L(f, s). It has a functional equation relating rf(z) to rf(−1Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of rf(z) lie on the circle ∣∣z∣∣=1/N‾‾√. We prove that these zeros are equidistributed when either k or N is large.