AGM aquariums and elliptic curves over arbitrary finite fields
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40993_2025_629_ReferencePDF.pdf
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Author(s)
Kayath, June
Lane, Connor
Neifeld, Ben
Ni, Tianyu
Xue, Hui
Date Issued
April 9, 2025
Journal
Research in Number Theory
Publisher
Springer International Publishing
Citation
Kayath, J., Lane, C., Neifeld, B. et al. AGM aquariums and elliptic curves over arbitrary finite fields. Res. number theory 11, 48 (2025).
Version
Author's final manuscript
Abstract
In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields F q , and study the resulting AGM graph with points ( a , b ) ∈ F q × F q and directed edges between points (a, b), ( a + b 2 , ab ) and (a, b), ( a + b 2 , - ab ) . The points in this graph are naturally associated to elliptic curves over 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s40993-025-00629-7
