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

TL;DR

This paper studies prime trees linked to prime subsets, establishing conditions for their finiteness, calculating their density, and applying these results to express cusp forms as products of Eisenstein series in certain modular forms settings.

Contribution

It introduces prime trees associated with prime subsets, characterizes their finiteness, and applies these findings to modular forms and Eisenstein series representations.

Findings

Conditions for prime trees to be finite

Density calculations for finite-type prime subsets

Representation of cusp forms as Eisenstein series products

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.