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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.23096 |
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| _version_ | 1866916008415985664 |
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| author | Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa |
| author_facet | Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa |
| contents | 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(τ)=f\left(\fracτ{p}\right)\quad(τ\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 $j$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23096 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa Number Theory 11F03, 12J25, 13F25 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(τ)=f\left(\fracτ{p}\right)\quad(τ\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 $j$. |
| title | On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences |
| topic | Number Theory 11F03, 12J25, 13F25 |
| url | https://arxiv.org/abs/2604.23096 |