Higher moments for symmetric powers of modular forms
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914300622274560 |
|---|---|
| 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 |