MATH Seminar

Title: Zeros of Period polynomials for symmetric power L-functions
Seminar: Algebra and Number Theory
Speaker: Robert Dicks of Clemson University
Contact: Santiago Arango, santiago.arango@emory.edu
Date: 2024-11-07 at 4:00PM
Venue: MSC E406
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Abstract: Suppose that $k$ and $N$ are positive integers. Let $f$ be a normalized cuspidal Hecke eigenform on $\Gamma_0(N)$ 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 $\{z \in \mathbb{C} : \|z\|=1/\sqrt{N}\}$.\\ \\ In this paper, we define a natural analogue of period polynomials for the symmetric power $L$-functions of $f$ and prove the corresponding Riemann hypothesis when $k$ is large enough.\\ \\ This is joint work with Hui Xue.