Dimension sequences of modular forms

TL;DR

This paper explores the possible dimensions of spaces of cuspidal modular forms of weight 2 across levels, generalizing a conjecture about their sequence covering all natural numbers and providing characterizations for various cases.

Contribution

It offers a comprehensive analysis and characterization of when the dimension sequences of modular form spaces can attain all natural numbers or other specified properties.

Findings

Characterization of when dimension sequences cover all natural numbers

Generalizations of Martin's conjecture to various modular form spaces

Complete classification of properties of dimension sequences

Abstract

For $N \geq 1$, let $S_{2}^{\text{new}}(N)$ denote the newspace of cuspidal modular forms of weight $2$ and level $N$. In 2004, Greg Martin conjectured that as a sequence in $N$, $\dim S_2^{\text{new}}(N)$ takes on all possible natural numbers. In this paper, we investigate several generalizations and variations of this type of problem. In each case, we provide a complete characterization of when such a property holds.