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The Klein-Vélu septic, 2-Division, and modular function fields of level 14

  • Masanobu Kaneko
  • Masato Kuwata
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Abstract

We study the 2-division of the Klein-Vélu elliptic normal septic. Its three non-zero 2-torsion points give rise to an explicit cubic equation over the function field of X(7). We identify the three roots of this cubic with modular functions expressed in terms of septic theta functions at τ/2,τ, and 2τ, and show that its splitting field is precisely the function field of X(14). This gives an explicit link between the 2-division geometry of the Klein-Vélu septic and the passage from full level 7 to full level $14$. Several intermediate modular function fields in the resulting S3-extension are also described.

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Cite

BibTeXRIS
BibTeX
@misc{kaneko2026kleinvélu,
  title = {{The Klein-Vélu septic, 2-Division, and modular function fields of level 14}},
  author = {Masanobu Kaneko and Masato Kuwata},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03455},
  eprint = {2609.03455},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03455},
}
RIS
TY  - UNPB
TI  - The Klein-Vélu septic, 2-Division, and modular function fields of level 14
AU  - Masanobu Kaneko
AU  - Masato Kuwata
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03455
UR  - https://arxiv.org/abs/2609.03455
AB  - We study the 2-division of the Klein-Vélu elliptic normal septic. Its three non-zero 2-torsion points give rise to an explicit cubic equation over the function field of X(7). We identify the three roots of this cubic with modular functions expressed in terms of septic theta functions at τ/2,τ, and 2τ, and show that its splitting field is precisely the function field of X(14). This gives an explicit link between the 2-division geometry of the Klein-Vélu septic and the passage from full level 7 to full level $14$. Several intermediate modular function fields in the resulting S_3-extension are also described.
ER  -