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arXiv · 2512.21088

Explicit constructions of cyclic N-isogenies

Daeyeol Jeon·Yongjae Kwon
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Abstract

The modular curve X_0(N) parametrizes elliptic curves together with a cyclic subgroup of order N, and hence cyclic N-isogenies. While explicit moduli descriptions of X_1(N) are well developed, a comparable construction for X_0(N) has remained incomplete. We give a uniform method for constructing explicit generators of C(X_0(N)), extending an approach of Dowd, and use them to obtain a concrete moduli interpretation of cyclic N-isogenies. This yields explicit formulas for sporadic rational points on X_0(N) and the associated isogenies, providing a unified solution to the moduli problem for X_0(N).

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Daeyeol Jeon, Yongjae Kwon. 2025-12-24. Explicit constructions of cyclic N-isogenies. https://arxiv.org/abs/2512.21088

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