Linear independence of periods for the symmetric square $L$-functions

TL;DR

This paper studies the linear independence of periods related to symmetric square $L$-functions for cusp forms, establishing conditions under which multiple such periods are linearly independent as the weight grows.

Contribution

It introduces periods associated with symmetric square $L$-functions and proves their linear independence for large weights, extending understanding of period relations.

Findings

Linear independence of $n$ periods for large $k$

Existence of up to $rac{ ext{log }k}{4}$ independent periods for $k o ext{large}$

Conditions relating weight size to period independence

Abstract

For $S_k$, the space of cusp forms of weight $k$ for the full modular group, we first introduce periods on $S_k$ associated to symmetric square $L$-functions. We then prove that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$, then any $n$ such periods are linearly independent. With some extra assumption, we also prove that for $k\geq e^{12}$, we can always pick up to $\frac{\log k}{4}$ arbitrary linearly independent periods.