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

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.

💡 Research Summary

The paper introduces a novel combinatorial object called the Additive Prime Tree (APT) associated with a finite set P of prime numbers. Starting from a pair of positive integers (x, y), the tree is generated by repeatedly applying the operations (X, Y) → (X, X+Y) and (X+Y, Y), but only when the current coordinates contain distinct primes from P: one prime must divide X and another (possibly the same) must divide Y. This rule forces the tree to be a subset of ℕ² and gives it a natural representation via the semigroup generated by the matrices L =