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arXiv · 2410.17969

AGM aquariums and elliptic curves over arbitrary finite fields

June Kayath·Connor Lane·Ben Neifeld·Tianyu Ni·Hui Xue
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Abstract

In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields $\mathbb{F}_q$, and study the resulting AGM graph with points $(a,b) \in \mathbb{F}_q \times \mathbb{F}_q$ and directed edges between points $(a,b)$, $(\frac{a+b}{2},\sqrt{ab})$ and $(a,b)$, $(\frac{a+b}{2},-\sqrt{ab})$. The points in this graph are naturally associated to elliptic curves over $\mathbb{F}_q$ in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph.

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June Kayath, Connor Lane, Ben Neifeld, Tianyu Ni, Hui Xue. 2024-10-23. AGM aquariums and elliptic curves over arbitrary finite fields. https://arxiv.org/abs/2410.17969

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