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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.01571 |
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| _version_ | 1866914300622274560 |
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| author | Yang, Jiong Yang, Zhishan |
| author_facet | Yang, Jiong Yang, Zhishan |
| contents | Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $λ_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq x}λ^l_{\Sym^d f}(n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher moments for symmetric powers of modular forms Yang, Jiong Yang, Zhishan Number Theory Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $λ_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq x}λ^l_{\Sym^d f}(n)$ 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. |
| title | Higher moments for symmetric powers of modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.01571 |