On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences

TL;DR

This paper proves a new integrality property of certain modular functions related to Kronecker-type congruences, extending classical results for the elliptic modular function j.

Contribution

It generalizes the classical Kronecker congruence relation for the modular function j to a broader class of meromorphic modular functions of level N.

Findings

If p ≡ ±1 mod N and f is integral over Z[j], then (1/p)(f_p^p - f)(f_p - f^p) is also integral over Z[j].

The result extends classical Kronecker congruences to more general modular functions.

The paper establishes conditions under which certain transformed modular functions maintain integrality.

Abstract

Let $N$ be a positive integer and let $f$ be a meromorphic modular function of level $N$ with rational Fourier coefficients. For a prime $p$, define a function $f_p$ on the complex upper half-plane $\mathbb{H}$ by \begin{equation*} f_p(\tau)=f\left(\frac{\tau}{p}\right)\quad(\tau\in\mathbb{H}). \end{equation*} Let $j$ be the elliptic modular function. We show that if $p\equiv 1$ or $-1\Mod{N}$ and $f$ is integral over $\mathbb{Z}[j]$, then \begin{equation*} \frac{1}{p}(f_p^p-f)(f_p-f^p) \end{equation*} is also integral over $\mathbb{Z}[j]$. This result generalizes the classical Kronecker congruence relation for $j$.