Divisibility Properties of Coefficients of Level $p$ Modular Functions for Genus Zero Primes

TL;DR

This paper extends Lehner's 1949 divisibility results for the $j$-invariant to a broader class of level $p$ modular functions, demonstrating high divisibility of their Fourier coefficients by primes 2, 3, 5, and 7.

Contribution

It introduces a canonical basis for level $p$ modular functions and proves their Fourier coefficients are often highly divisible by the primes 2, 3, 5, and 7.

Findings

Fourier coefficients exhibit high divisibility by primes 2, 3, 5, and 7.

Extension of Lehner's results to level $p$ modular functions.

Canonical basis constructed for these modular functions.

Abstract

Lehner's 1949 results on the $j$-invariant showed high divisibility of the function's coefficients by the primes $p\in\{2,3,5,7\}$. Expanding his results, we examine a canonical basis for the space of level $p$ modular functions holomorphic at the cusp 0. We show that the Fourier coefficients of these functions are often highly divisible by these same primes.