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

TL;DR

The paper introduces Additive Prime Trees (APT) associated with finite prime sets $P$ and establishes finiteness criteria and density bounds for finite-type $P$, enriching the combinatorial understanding of prime-based structures. It then connects this combinatorics to modular form theory, proving that for finite-type $P$ and levels $N = N'\prod_{p\in P} p^{a_p}$ with $a_p\ge2$, the space $\mathcal{M}_k(\Gamma_0(N))$ decomposes as $\mathcal{E}_k(\Gamma_0(N)) \oplus \mathcal{Q}_k(N)$, where $\mathcal{Q}_k(N)$ is generated by products of Eisenstein series, and the cuspidal/new subspace is captured by $\mathcal{S}_k^{\mathrm{new}}(\Gamma_0(N))$; this extends Dickson–Neururer's results to broader levels. The work provides a density estimate for finite-type sets, an algorithmic approach to enumerate primitive sets, and a framework linking APT structure to vanishing of twisted $L$-values via modular symbols. Overall, it advances the interplay between additive prime trees and the arithmetic of modular forms, offering new avenues to understand level decompositions and their prevalence.

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 $f \in \mathcal{S}_k(Γ_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type.

Theorems & Definitions (21)

• Theorem 1.1: DN
• Theorem 1.2: DN
• Definition 1.3
• Example 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1
• Proposition 2.2
• Example 2.3