Zeros of symmetric power period polynomials

2025/01/29 by Robert Dicks, Hui Xue, Dicks, Robert +1
Mathematics · #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.18024

openalex publication_date 2025/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that k and N are positive integers. Let f be a newform on Γ₀(N) of weight k with L-function L_f(s). Previous works have studied the zeros of the period polynomial r_f(z), which is a generating function for the critical values of L_f(s) and has a functional equation relating z and -1/Nz. In particular, r_f(z) satisfies a version of the Riemann hypothesis: all of its zeros are on the circle of symmetry \z ∈ \C : |z|=1/√(N)\. In this paper, for a positive integer m, we define a natural analogue of r_f(z) for the m^th symmetric power L-function of f when N is squarefree. Our analogue also has a functional equation relating z and -1/Nz. We prove the corresponding version of the Riemann hypothesis when k is large enough. Moreover, when k>2(log₂(13e^2π/9)+m)+1, we prove our result when N is large enough.

Related

• The "Riemann Hypothesis" is True for Period Polynomials of Almost All\n Newforms
• Zeros of even and odd period polynomials
• The Riemann Hypothesis for period polynomials of cusp forms
• The Riemann Hypothesis for period polynomials of Hilbert modular forms
• Zeta-polynomials for modular form periods