Explicit constructions of cyclic N-isogenies
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911337369567232 |
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| author | Jeon, Daeyeol Kwon, Yongjae |
| author_facet | Jeon, Daeyeol Kwon, Yongjae |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21088 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit constructions of cyclic N-isogenies Jeon, Daeyeol Kwon, Yongjae Number Theory 11G05, 11G18 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). |
| title | Explicit constructions of cyclic N-isogenies |
| topic | Number Theory 11G05, 11G18 |
| url | https://arxiv.org/abs/2512.21088 |