On the finiteness of prime trees and their relation to modular forms
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917235159728128 |
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| author | Fujiyoshi, Yusuke |
| author_facet | Fujiyoshi, Yusuke |
| contents | In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As an application, we show that for weight $k \ge 2$ and levels $N = N'\prod_{p \in P} p^{a_p}$, where $N'$ is squarefree and $a_{p} \geq 2$, every cusp form $f \in \mathcal{S}_k(Γ_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_23016 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the finiteness of prime trees and their relation to modular forms Fujiyoshi, Yusuke Number Theory 11F11, 11F33, 11A41 In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As an application, we show that for weight $k \ge 2$ and levels $N = N'\prod_{p \in P} p^{a_p}$, where $N'$ is squarefree and $a_{p} \geq 2$, every cusp form $f \in \mathcal{S}_k(Γ_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type. |
| title | On the finiteness of prime trees and their relation to modular forms |
| topic | Number Theory 11F11, 11F33, 11A41 |
| url | https://arxiv.org/abs/2601.23016 |