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PreprintCC BY 4.0arXiv

Strong Weil Degree Divisibility at Higher Levels

  • Daeyeol Jeon
  • Yongjae Kwon
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Abstract

Let πE:X0(M)→ E be the strong Weil parametrization with Manin constant cE. We prove °πE| cEΩ(N/M)°g for every multiple N of M and every nonconstant morphism g:X0(N)→ E' over ℚ, where E' is ℚ-isogenous to E and Ω counts prime factors with multiplicity. When cE=1, as is known for squarefree M, the modular degree at level M therefore divides every such degree at every higher level. As an application of the divisibility theorem, we prove that no X0(N)/ℚ admits a morphism over ℚ of positive odd degree at most 1645 to an elliptic curve of positive ℚ-rank. For a fixed target E' and a generator u:E→ E', we also prove that if the Manin constant cu∘πE=1, the old homomorphisms induced by degeneracy maps form an integral basis of Homℚ(J0(N),E'), and the old degree matrix determines the exact morphism degrees. The proofs bound denominators in the rational old basis. The divisibility and lattice results extend to compatible towers of intermediate modular curves, including the X1-tower.

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Cite

BibTeXRIS
BibTeX
@misc{jeon2026strong,
  title = {{Strong Weil Degree Divisibility at Higher Levels}},
  author = {Daeyeol Jeon and Yongjae Kwon},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.06054},
  eprint = {2608.06054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.06054},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Strong Weil Degree Divisibility at Higher Levels
AU  - Daeyeol Jeon
AU  - Yongjae Kwon
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.06054
UR  - https://arxiv.org/abs/2608.06054
AB  - Let π_E:X_0(M)→ E be the strong Weil parametrization with Manin constant c_E. We prove °π_E| c_E^Ω(N/M)°g for every multiple N of M and every nonconstant morphism g:X_0(N)→ E' over ℚ, where E' is ℚ-isogenous to E and Ω counts prime factors with multiplicity. When c_E=1, as is known for squarefree M, the modular degree at level M therefore divides every such degree at every higher level. As an application of the divisibility theorem, we prove that no X_0(N)/ℚ admits a morphism over ℚ of positive odd degree at most 1645 to an elliptic curve of positive ℚ-rank. For a fixed target E' and a generator u:E→ E', we also prove that if the Manin constant c_u∘π_E=1, the old homomorphisms induced by degeneracy maps form an integral basis of Hom_ℚ(J_0(N),E'), and the old degree matrix determines the exact morphism degrees. The proofs bound denominators in the rational old basis. The divisibility and lattice results extend to compatible towers of intermediate modular curves, including the X_1-tower.
ER  -