Hauptmoduln and even-order mock theta functions modulo 2

TL;DR

This paper explores the parity of Fourier coefficients of the elliptic modular j-function and their relation to mock theta functions and hauptmoduln modulo 2, revealing new parity phenomena and connections.

Contribution

It establishes a novel parity relationship between coefficients of the j-function, mock theta functions, and hauptmoduln modulo 2, extending understanding of their arithmetic properties.

Findings

Coefficients $c_1(8n-1)$ share parity with $c_{ u_2}(n)$ of $rac12;$2nd order mock theta function.

Parity phenomena also observed among various hauptmoduln.

The study links the parity of Fourier coefficients to mock theta functions and hauptmoduln modulo 2.

Abstract

The Fourier coefficients $c_1(n)$ of the elliptic modular $j$-function are always even for $n \not\equiv 7 \pmod{8}$. In contrast, for $n \equiv 7 \pmod{8}$, it is conjectured that ``half" of the coefficients take odd values. In this article, we first observe in detail when $c_1(8n-1)$ is odd and show that the coefficients share the same parity as the coefficients $c_{\mu_2}(n)$ of the 2nd order mock theta function $\mu_2(q)$. Furthermore, we prove that this phenomenon also holds among several hauptmoduln and between hauptmoduln and even-order mock theta functions.