Unimodularity of zeros of period polynomials of Hecke eigenforms

Abstract

We prove that all the zeros of the full period function of any Hecke eigenform lie on the unit circle |z| = 1.

Riemann hypothesis for period polynomials of modular forms

“…Here we consider larger weights by reformulating the previous approach of [11] and [10]. Recast the definition (1.5) of P f (z) as…”

“…so that if ρ is a zero of r f (z) then so is −1/(Nρ). In analogy with the Riemann hypothesis, we may ask whether all the zeros of r f (z) lie on the circle |ρ| = 1/ √ N. For Hecke eigenforms on SL 2 (Z), this was recently established by El-Guindy and Raji [10], who showed that the zeros of r f (z) (for N = 1) are all on the unit circle |z| = 1. Their work was inspired by the previous work by Conrey, Farmer and Imamoḡlu [11], who proved an analogous result for odd period polynomials again for full level.…”

On a secant Dirichlet series and Eichler integrals of Eisenstein series

“…We again find that, at least conjecturally, most of the roots lie on a circle in the complex plane. The observations suggest that the problem solved by [CFI12] and [EGR13] is interesting in the higher level case as well.…”

Period polynomials, derivatives of L-functions, and zeros of polynomials

“…However, in this case, there are no trivial zeros, and all zeros of r f lie on the unit circle. This is summarized in the following result, shown by El-Guindy and Raji in [14] for level 1 and for general level N by Jin et al in [20]. Remark 2.8 Explicit approximations for the exact locations of the zeroes were given in [20].…”

“…This allows for a more convenient investigation of the location of the zeros thanks to the following result. Lemma 2.9 Theorem 2.2 of [14]) If h(z) is a polynomial of degree n with all zeros inside the unit disk |z| ≤ 1, then for any d ≥ n and λ on the unit circle, the polynomial…”