If f is a level-N meromorphic modular function with rational coefficients that is integral over Z[j] and p ≡ ±1 mod N, then (1/p)(f_p^p - f)(f_p - f^p) is integral over Z[j], generalizing Kronecker's congruence for j.
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On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences
If f is a level-N meromorphic modular function with rational coefficients that is integral over Z[j] and p ≡ ±1 mod N, then (1/p)(f_p^p - f)(f_p - f^p) is integral over Z[j], generalizing Kronecker's congruence for j.