Dynamical structure of AGM over finite fields with order congruent to $$5 \bmod 8$$

Abstract Motivated by classical works of Gauss and Euler on the arithmetic–geometric mean (AGM), Ono and his collaborators (Am Math Mon 130(4):355–369, 2023; Contemp Math 818:197–210, 2025) investigated the union of AGM sequences over finite fields $${{\mathbb {F}}}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> , where $$q \equiv 3 \bmod 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>3</mml:mn> <mml:mspace/> <mml:mo>mod</mml:mo> <mml:mspace/> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , which they refer to as swarms of jellyfish. A recent paper (Kayath et al. in Res Number Theory 11(48): 2025) extends some of their results to all finite fields with odd characteristic. For $$q \equiv 5 \bmod 8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>5</mml:mn> <mml:mspace/> <mml:mo>mod</mml:mo> <mml:mspace/> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> , we reveal finer details about the structure of the connected components, which turn out to be variants of jellyfish with longer and branched tentacles. Moreover, we determine the total population of these swarms in terms of the celebrated base “congruent number” elliptic curve.