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

Tianyu Ni; Hui Xue

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arXiv · 2507.17608

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

Tianyu Ni·Hui Xue

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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.

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Tianyu Ni, Hui Xue. 2025-07-23. Linear independence of periods for the symmetric square $L$-functions. https://arxiv.org/abs/2507.17608

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