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Cubic AGM graphs and Hessian $3$-isogenies over finite fields 𝔽q of odd characteristic with q ≑ 2 pmod3

  • Yuji Hashimoto
  • Koji Nuida
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Abstract

Let 𝔽q be a finite field of odd characteristic with q ≑ 2 pmod3. We study the directed graph defined by the Borwein--Borwein cubic arithmetic--geometric mean (AGM) over 𝔽q. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also show that each edge corresponds to a $3$-isogeny defined over 𝔽q between the curves associated with its initial and terminal vertices. We then use a counting formula for Hessian curves to derive a lower bound for the number of cycles.

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Cite

BibTeXRIS
BibTeX
@misc{hashimoto2026cubic,
  title = {{Cubic AGM graphs and Hessian $3$-isogenies over finite fields $\mathbbF_q$ of odd characteristic with $q \equiv 2 \pmod3$}},
  author = {Yuji Hashimoto and Koji Nuida},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14300},
  eprint = {2609.14300},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14300},
}
RIS
TY  - UNPB
TI  - Cubic AGM graphs and Hessian $3$-isogenies over finite fields 𝔽_q of odd characteristic with q ≑ 2 pmod3
AU  - Yuji Hashimoto
AU  - Koji Nuida
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14300
UR  - https://arxiv.org/abs/2609.14300
AB  - Let 𝔽_q be a finite field of odd characteristic with q ≑ 2 pmod3. We study the directed graph defined by the Borwein--Borwein cubic arithmetic--geometric mean (AGM) over 𝔽_q. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also show that each edge corresponds to a $3$-isogeny defined over 𝔽_q between the curves associated with its initial and terminal vertices. We then use a counting formula for Hessian curves to derive a lower bound for the number of cycles.
ER  -