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arxiv: 2604.23096 · v2 · submitted 2026-04-25 · 🧮 math.NT

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

Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin This is my paper

Pith reviewed 2026-05-14 21:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords modular functionsintegrality over Z[j]Kronecker congruencelevel Nelliptic modular functionFourier coefficientsmeromorphic modular functions
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The pith

If a level-N modular function f with rational coefficients is integral over Z[j], then for primes p congruent to 1 or -1 mod N the expression (1/p)(f_p^p - f)(f_p - f^p) is also integral over Z[j].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that integrality over Z[j] is preserved under a specific transformation for meromorphic modular functions of level N. When p satisfies p ≡ 1 or -1 mod N, the averaged expression built from f and its scaled version f_p remains integral over Z[j]. This extends the classical Kronecker congruence, which is recovered when f equals j itself. A reader cares because the result supplies a systematic way to produce new integral modular functions while controlling their arithmetic behavior at the level of the j-invariant.

Core claim

Let N be a positive integer and let f be a meromorphic modular function of level N with rational Fourier coefficients. Define f_p(τ) = f(τ/p) for τ in the upper half-plane. The paper shows that if p ≡ 1 or −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]. This generalizes the classical Kronecker congruence relation for j.

What carries the argument

The map that sends f to (1/p)(f_p^p − f)(f_p − f^p), where f_p is obtained by scaling the argument of f by the prime p; this map preserves integrality over Z[j] precisely when p ≡ ±1 mod N.

If this is right

  • The classical Kronecker congruence for the j-function is recovered as the special case f = j.
  • New functions that are integral over Z[j] can be constructed iteratively from known ones by repeated application of the map.
  • The result applies uniformly to every meromorphic modular function of exact level N with rational Fourier coefficients that starts integral over Z[j].
  • The integrality statement is stable under the Galois action on the Fourier coefficients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Iterated application may generate an ascending chain of integral functions whose fields of definition stabilize at a fixed level.
  • The construction suggests that the integral closure of Z[j] inside the field of level-N functions is closed under this averaged p-power map.
  • Explicit computation for small N and small integral f could produce new examples of integral bases over Z[j] that are not obviously related to classical generators.

Load-bearing premise

f must be integral over Z[j] and the prime p must be congruent to 1 or −1 modulo the level N of f; drop either condition and the integrality conclusion no longer holds.

What would settle it

Take any explicit integral modular function f of level N (for example a suitable multiple of a Weber function), choose a prime p not congruent to ±1 mod N, expand the expression (1/p)(f_p^p − f)(f_p − f^p) as a Laurent series in q = e^{2πiτ} and check whether it remains integral over Z[j] or acquires a denominator divisible by p.

read the original 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 $j$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves that if f is a meromorphic modular function of exact level N with rational Fourier coefficients that is integral over Z[j], and p is a prime with p ≡ ±1 mod N, then the function (1/p)(f_p^p − f)(f_p − f^p), where f_p(τ) = f(τ/p), is also integral over Z[j]. This is presented as a direct generalization of the classical Kronecker congruence relation satisfied by the j-invariant itself.

Significance. If the central claim holds, the result supplies a uniform integrality statement for a natural class of level-N modular functions under the stated congruence condition on p. This could streamline arguments involving integral models of modular functions and their reductions modulo primes, extending tools from the theory of complex multiplication beyond the j-function alone. The absence of free parameters or ad-hoc constants in the statement is a positive feature.

major comments (2)
  1. [§3] §3, main theorem: the argument that the given expression remains integral over Z[j] appears to reduce the problem to the q-expansion principle at the cusp ∞, but the manuscript does not explicitly verify that the denominator p is canceled after applying the monic polynomial relation satisfied by f; a concrete expansion of the first few terms of the resulting q-series would make the cancellation transparent.
  2. [§2.2] §2.2, definition of f_p: while f_p(τ) = f(τ/p) is well-defined on H, the transformation law under the level-N group must be checked to confirm that f_p remains a modular function of level dividing Np; the paper invokes p ≡ ±1 mod N to restore level N, but the precise matrix representatives used in the proof are not listed.
minor comments (2)
  1. [Introduction] The abstract and introduction both state the result for 'rational Fourier coefficients,' yet the proof sketch in §3 only uses the q-expansion at ∞; it would be helpful to note whether the same integrality holds at other cusps.
  2. Notation: the symbol f_p is introduced in the abstract but the subscript p is reused for the prime; a brief remark distinguishing the two uses would avoid confusion for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§3] §3, main theorem: the argument that the given expression remains integral over Z[j] appears to reduce the problem to the q-expansion principle at the cusp ∞, but the manuscript does not explicitly verify that the denominator p is canceled after applying the monic polynomial relation satisfied by f; a concrete expansion of the first few terms of the resulting q-series would make the cancellation transparent.

    Authors: We agree that an explicit verification of the p-cancellation would strengthen the exposition. Although the monic integrality relation for f over Z[j] combined with the q-expansion principle at infinity implies that the full expression is integral over Z[j], we will add in the revised manuscript a direct computation of the first few coefficients in the q-expansion of (1/p)(f_p^p - f)(f_p - f^p) to exhibit the cancellation explicitly. revision: yes

  2. Referee: [§2.2] §2.2, definition of f_p: while f_p(τ) = f(τ/p) is well-defined on H, the transformation law under the level-N group must be checked to confirm that f_p remains a modular function of level dividing Np; the paper invokes p ≡ ±1 mod N to restore level N, but the precise matrix representatives used in the proof are not listed.

    Authors: We thank the referee for this observation. The congruence p ≡ ±1 (mod N) guarantees that conjugation by the matrix (p 0; 0 1) maps the level-N congruence subgroup into itself up to the action of SL_2(Z), thereby preserving modularity at level N. In the revised version we will explicitly list the relevant matrix representatives (including the standard generators of Γ(N) and the action of the Atkin-Lehner-type element) and verify the transformation law for f_p step by step. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no reduction to inputs by construction

full rationale

The paper proves a generalization of the classical Kronecker congruence for j to meromorphic modular functions f of level N that are integral over Z[j], under the condition p ≡ ±1 mod N. The statement and the expression (1/p)(f_p^p - f)(f_p - f^p) follow directly from the transformation law f_p(τ) = f(τ/p) together with the integrality hypothesis and rational Fourier coefficients; no step equates the conclusion to a fitted parameter, self-referential definition, or load-bearing self-citation. The argument is independent of the present authors' prior results and rests on standard q-expansion and modular transformation properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background facts from the theory of modular functions and their integrality over Z[j]; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Meromorphic modular functions of level N with rational Fourier coefficients are integral over Z[j] when they satisfy a monic polynomial equation with coefficients in Z[j]
    This is the hypothesis of the theorem and is taken as given from classical theory.
  • domain assumption The scaled function f_p(τ) = f(τ/p) preserves the relevant transformation properties under the stated congruence condition on p
    Invoked in the definition of f_p and the conclusion.

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