A Kronecker congruence relation for modular functions of higher level and genus

Abstract

Let j be the elliptic modular function, a weakly holomorphic modular function for SL2(Z). Weber showed that for each prime p the modular polynomial 'T'p(x, y) of j satisfies what is known as the Kronecker congruence relation 'T'p(x, y) equivalent to (xp-y)(x-yp) (mod pZ[x, y]). We give a generalization of this congruence applicable to certain weakly holomorphic modular functions of higher level in terms of integrality over Z[j]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.