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arXiv · 2601.23016

On the finiteness of prime trees and their relation to modular forms

Yusuke Fujiyoshi
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Abstract

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 $f \in \mathcal{S}_k(\Gamma_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type.

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Yusuke Fujiyoshi. 2026-01-30. On the finiteness of prime trees and their relation to modular forms. https://arxiv.org/abs/2601.23016

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