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

The paper generalizes the classical Kronecker congruence relation by proving that for a modular function $f$ of level $N$ that is integral over $\mathbb{Z}[j]$ , the expression $\frac{1}{p}(f_p^p-f)(f_p-f^p)$ remains integral over $\mathbb{Z}[j]$ whenever $p \equiv \pm 1 \pmod{N}$ .

Original authors: Ho Yun Jung, Ja Kyung Koo, Dong Hwa Shin

Published 2026-04-28

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CC BY 4.0

Original authors: Ho Yun Jung, Ja Kyung Koo, Dong Hwa Shin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master locksmith. You have a collection of incredibly complex, high-security locks (these are the modular functions). These locks are so intricate that they don't just work with one key; they work with entire patterns of keys.

The mathematicians in this paper are studying a very specific, legendary set of locks called the $j$ -function. This $j$ -function is like the "Master Lock" of the mathematical universe. Almost every other complex lock can be described or "built" using pieces of this Master Lock.

Here is the breakdown of their discovery using a few metaphors:

1. The "DNA" of the Locks (Fourier Coefficients)

Every complex lock has a blueprint—a list of tiny, microscopic measurements that tell you exactly how it’s shaped. In math, we call these measurements Fourier coefficients.

Usually, these measurements can be any messy, complicated number (like $1/7$ or $\sqrt{2}/3$ ). However, the authors are interested in a special class of "clean" locks. These are locks where the measurements are integers (whole numbers like $1, 2, 3$). If a lock is "integral over $\mathbb{Z}[j]$ ," it means its blueprint is built entirely out of the "DNA" of the Master Lock.

2. The "Mirror Test" (Kronecker Congruence)

The paper focuses on a phenomenon called the Kronecker congruence.

Imagine you take one of these complex locks and you look at it through a special "Prime Mirror" (a prime number $p$ ). When you look through this mirror, the lock should look almost exactly like a "ghost" version of itself.

Specifically, if you take the lock, perform a certain mathematical transformation (like stretching it or rotating it), and then compare it to the original lock, the difference between them should be "invisible" to the mirror. In math terms, the difference is divisible by $p$ . It’s like saying if you take a high-resolution photo of a face and a slightly blurry one, and you look at them through a very specific filter, the two photos become indistinguishable.

3. The Big Discovery: Generalizing the Rule

For a long time, mathematicians knew this "Mirror Test" worked for the Master Lock ( $j$ ). They also suspected it might work for other, more complicated locks, but they couldn't prove it for all of them. They had a "conjecture"—a highly educated guess.

The authors of this paper proved the guess was right.

They showed that if you have a complex lock that is built from the Master Lock's DNA, and you pass it through this "Prime Mirror," the resulting "ghost" patterns will always align perfectly. They proved that a specific mathematical formula (the one involving $F$ in the paper) always results in a "clean" lock with whole-number measurements.

Summary in Plain English

The Problem: We knew the "Master Lock" ( $j$ ) had a special property where it looked identical to its "ghost version" when viewed through a prime-number filter. We thought other complex locks built from this Master Lock might have the same property, but we couldn't prove it.

The Solution: These authors used advanced "valuation" techniques (think of this as a high-powered microscope that measures how "divisible" a number is) to prove that this property is universal. If a function is built from the Master Lock, it will always obey this beautiful, symmetrical rule when viewed through the lens of a prime number.

Why it matters: It confirms that there is a deep, hidden order in the world of modular functions. It shows that the "DNA" of the Master Lock carries its special symmetry properties down to all the more complex structures built from it.

This paper, titled "On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences" by Jung, Koo, and Shin, addresses a central problem in the arithmetic theory of modular functions: determining when certain combinations of modular functions are integral over the ring $\mathbb{Z}[j]$ .

1. The Problem

The study of modular functions often centers on the elliptic modular function $j(\tau)$ . A classical result by Weber involves the modular polynomial $\Phi_p(x, y)$ , which satisfies the Kronecker congruence relation:
$\Phi_p(x, y) \equiv (x^p - y)(x - y^p) \pmod{p\mathbb{Z}[x, y]}$
While this is well-understood for $j$ , the authors seek to generalize this phenomenon to modular functions of higher level $N$ . Specifically, they investigate a conjecture made by Jung et al. regarding the integrality of a specific function $F$ constructed from a modular function $f$ of level $N$ and its "scaled" version $f_p(\tau) = f(\tau/p)$ .

2. Methodology

The authors depart from previous approaches that relied heavily on class field theory and the study of singular values (which often only yield results after taking products over Galois conjugates). Instead, they employ a direct analytic approach based on:

Fourier Expansion Analysis: They examine the $q$ -expansions of modular functions to determine integrality.
Exponential Valuations: They introduce a specialized valuation $w_P$ on the field of meromorphic modular functions $F_M$ . This valuation measures the "size" of the denominators of the Fourier coefficients with respect to a prime ideal $P$ .
Integrality Criterion: They establish a rigorous criterion (Proposition 6.2) stating that a function $g \in F_M$ is integral over $\mathbb{Z}[j]$ if and only if $g$ is weakly holomorphic and all its Galois conjugates $g \circ \alpha$ (for $\alpha \in SL_2(\mathbb{Z})$ ) have Fourier coefficients that are algebraic integers.
$p$ -th Power Arithmetic: They utilize the properties of the Frobenius endomorphism on power series rings over finite fields to handle the $p$ -th power congruences.

3. Key Contributions and Results

The paper's primary achievement is the proof of the conjecture regarding the integrality of the generalized Kronecker function.

Main Theorem (Theorem 8.1):
Let $f$ be a modular function of level $N$ that is integral over $\mathbb{Z}[j]$ . For any prime $p$ such that $p \equiv \pm 1 \pmod N$ , the function:
$F = \frac{1}{p} \left( f_p^p - f \begin{pmatrix} 1 & 0 \\ 0 & p \end{pmatrix} \right) \left( f_p - \left( f \begin{pmatrix} 1 & 0 \\ 0 & p \end{pmatrix} \right)^p \right)$
is integral over $\mathbb{Z}[j]$ . This leads to the Kronecker-type congruence:
$\left( f_p^p - f \begin{pmatrix} 1 & 0 \\ 0 & p \end{pmatrix} \right) \left( f_p - \left( f \begin{pmatrix} 1 & 0 \\ 0 & p \end{pmatrix} \right)^p \right) \equiv 0 \pmod{p\mathcal{O}_{F_N^p}}$

Additional Results:

Corollary 8.2: Provides a dual version of the function $G(\tau)$ involving $f(\tau)^p$ and $f(p\tau)$ .
Corollary 8.5 (Complex Multiplication): They apply the result to the theory of Complex Multiplication (CM), showing that for an imaginary quadratic field $K$ , the values of these functions at CM points satisfy specific congruences modulo $p\mathcal{O}_{K(N_p)}$ .

4. Significance

The significance of this work is twofold:

Theoretical Generalization: It successfully extends the classical Kronecker congruence from the level 1 case ( $j$ -function) to arbitrary levels $N$ , provided the prime $p$ satisfies certain congruence conditions.
Methodological Shift: By proving the result through direct $q$ -expansion and valuation-theoretic methods rather than class field theory, the authors provide a more "intrinsic" proof that works for individual functions rather than just their products over Galois orbits. This provides a more granular tool for researchers working in the arithmetic of modular curves.