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PreprintCC BY 4.0arXiv

D-elliptic modular curves X0(N)

  • Maarten Derickx
  • Daeyeol Jeon
  • Yongjae Kwon
  • Petar Orlić
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Abstract

We present a method for determining whether there exists a degree D rational map from X0(N) to some elliptic curve E/ℚ. Moreover, we explain how to obtain a quadratic form that represents all possible degrees of such maps using the degree pairing method. This method previously required odd analytic rank and some additional technical conditions to be satisfied. In this paper, we extended the method to even rank curves as well and remove all technical conditions. We also show how to prove or disprove the existence of degree D maps to elliptic curves by a lifting criterion on the lattices H1(−,ℤ) As an application of this method, we determine all D-elliptic curves X0(N) for all D≤ 100. Interestingly, in some cases we found degree D rational maps that we could not explain by degeneracy maps, quotient maps, and modular parameterisation.

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Cite

BibTeXRIS
BibTeX
@misc{derickx2026delliptic,
  title = {{$D$-elliptic modular curves $X_0(N)$}},
  author = {Maarten Derickx and Daeyeol Jeon and Yongjae Kwon and Petar Orlić},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24363},
  eprint = {2609.24363},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24363},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - D-elliptic modular curves X_0(N)
AU  - Maarten Derickx
AU  - Daeyeol Jeon
AU  - Yongjae Kwon
AU  - Petar Orlić
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24363
UR  - https://arxiv.org/abs/2609.24363
AB  - We present a method for determining whether there exists a degree D rational map from X_0(N) to some elliptic curve E/ℚ. Moreover, we explain how to obtain a quadratic form that represents all possible degrees of such maps using the degree pairing method. This method previously required odd analytic rank and some additional technical conditions to be satisfied. In this paper, we extended the method to even rank curves as well and remove all technical conditions. We also show how to prove or disprove the existence of degree D maps to elliptic curves by a lifting criterion on the lattices H_1(−,ℤ) As an application of this method, we determine all D-elliptic curves X_0(N) for all D≤ 100. Interestingly, in some cases we found degree D rational maps that we could not explain by degeneracy maps, quotient maps, and modular parameterisation.
ER  -