Higher moments for symmetric powers of modular forms

TL;DR

This paper derives asymptotic formulas for moments of symmetric power Fourier coefficients of modular forms and quadratic form values, extending previous results to all degrees and powers using Galois representations and $L$-functions.

Contribution

It provides the first comprehensive asymptotic formulas for moments of symmetric power coefficients for all degrees and powers, improving upon prior limited cases.

Findings

Established asymptotic formulas for moments of symmetric power coefficients.

Derived asymptotics for sums over quadratic form values.

Generalized previous results to all degrees and powers.

Abstract

Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $\lambda_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq x}\lambda^l_{\Sym^d f}(n)$ for all positive integers $d$ and $l$. We also prove an asymptotic formula for the corresponding sum over the values of any positive definite binary quadratic form $Q$. Our results generalize and improve upon previous work, which was limited to small values of $d$ or $l$. The proofs rely on the decomposition of $\ell$-adic Galois representations and the analytic properties of the associated $L$-functions.