Bounds for moments of symmetric square $L$-functions

TL;DR

This paper establishes sharp lower bounds for the moments of symmetric square L-functions at the central point for a family of holomorphic cusp forms, with results depending on the value of k and assumptions like GRH.

Contribution

It provides the first sharp lower bounds for all real k ≥ 1/2 unconditionally and extends bounds under GRH for other ranges of k.

Findings

Unconditional sharp lower bounds for k ≥ 1/2.

Conditional sharp bounds for 0 ≤ k < 1/2.

Sharp upper bounds for k ≥ 0 under GRH.

Abstract

We study the $2k$-th moment at the central point of the family of symmetric square $L$-functions attached to holomorphic Hecke cusp forms of level one, weight $\kappa$. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally. Assuming the truth of the generalized Riemann hypothesis, we also obtain sharp lower bounds for all real $0 \leq k < 1/2$ and sharp upper bounds for all real $k \geq 0$.