Congruence classes for modular forms over small sets
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914741023145984 |
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| author | Bhakta, Subham Krishnamoorthy, S. Muneeswaran, R. |
| author_facet | Bhakta, Subham Krishnamoorthy, S. Muneeswaran, R. |
| contents | J.P. Serre showed that for any integer $m,~a(n)\equiv 0 \pmod m$ for almost all $n,$ where $a(n)$ is the $n^{\text{th}}$ Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of cuspforms and study $\#\{a(n) \pmod m\}_{n\leq x}$ over the set of integers with $O(1)$ many prime factors. Moreover, we show that any residue class $a\in \mathbb{Z}/m\mathbb{Z}$ can be written as the sum of at most thirteen Fourier coefficients, which are polynomially bounded as a function of $m.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02725 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Congruence classes for modular forms over small sets Bhakta, Subham Krishnamoorthy, S. Muneeswaran, R. Number Theory Primary 11F30, 11L07, Secondary 11P05, 11B37, 11F80 J.P. Serre showed that for any integer $m,~a(n)\equiv 0 \pmod m$ for almost all $n,$ where $a(n)$ is the $n^{\text{th}}$ Fourier coefficient of any modular form with rational coefficients. In this article, we consider a certain class of cuspforms and study $\#\{a(n) \pmod m\}_{n\leq x}$ over the set of integers with $O(1)$ many prime factors. Moreover, we show that any residue class $a\in \mathbb{Z}/m\mathbb{Z}$ can be written as the sum of at most thirteen Fourier coefficients, which are polynomially bounded as a function of $m.$ |
| title | Congruence classes for modular forms over small sets |
| topic | Number Theory Primary 11F30, 11L07, Secondary 11P05, 11B37, 11F80 |
| url | https://arxiv.org/abs/2302.02725 |