Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

TL;DR

This paper investigates the symmetry type of low-lying zeros of symmetric power L-functions weighted by symmetric square L-values in the level aspect, revealing a $z$-interpolation of known results and proposing a new conjecture.

Contribution

It determines the symmetry type of low-lying zeros for symmetric power L-functions weighted by special symmetric square L-values, extending previous results with a novel $z$-interpolation.

Findings

Weighted density matches known symmetry types for 0<z≤1

At z=0, density differs only when r=2, not aligning with predictions

Proposes a conjecture on weighted density of low-lying zeros

Abstract

For a totally real number field $F$ and its ad\`ele ring $\mathbb{A}_F$, let $\pi$ vary in the set of irreducible cuspidal automorphic representations of ${\rm PGL}_2(\mathbb{A}_F)$ corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power $L$-functions $L(s,{\rm Sym}^r(\pi))$ weighted by special values of symmetric square $L$-functions $L(\frac{z+1}{2},{\rm Sym}^2(\pi))$ at $z \in [0, 1]$ in the level aspect. If $0 < z \le 1$, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power $L$-functions for $F=\mathbb{Q}$ with harmonic weight. Hence our result is regarded as a $z$-interpolation of Ricotta and Royer's result. If $z=0$, density of low-lying zeros weighted by central values is a different type only when $r=2$, and it does not appear in random matrix theory as Katz and Sarnak predicted. Moreover, we propose a conjecture on weighted density of low-lying zeros of $L$-functions by special $L$-values. In the latter part, Appendices A, B and C are dedicated to the comparison among several generalizations of Zagier's parameterized trace formula. We prove that the explicit Jacquet-Zagier type trace formula (the ST trace formula) by Tsuzuki and the author recovers all of Zagier's, Takase's and Mizumoto's formulas by specializing several data. Such comparison is not so straightforward and includes non-trivial analytic evaluations.