Hypergeometry and the AGM over Finite Fields

TL;DR

This paper explores finite field analogues of the arithmetic-geometric mean using hypergeometric functions, revealing new identities for class numbers and connections to elliptic curves over finite fields.

Contribution

It introduces a finite field analogue of AGM, studies associated graph structures called jellyfish swarms, and links these to class numbers and elliptic curves.

Findings

New identities for Gauss' class numbers of quadratic forms

Jellyfish sizes relate to prime order in class groups

Finite field hypergeometric functions organize elliptic curves

Abstract

One of the most celebrated applications of Gauss' $_2F_1$ hypergeometric functions is in connection with the rapid convergence of sequences and special values that arise in the theory of arithmetic and geometric means. This theory was the inspiration for a recent paper \cite{jelly1} in which a finite field analogue of AGM$_\mathbb{R}$ was defined and then studied using finite field hypergeometric functions. Instead of convergent sequences, one gets directed graphs that combine to form disjoint unions of graphs that individually resemble "jellyfish". Echoing the connection of hypergeometric functions to periods of elliptic curves, these graphs organize elliptic curves over finite fields. Here we use such "jellyfish swarms" to prove new identities for Gauss' class numbers of positive definite binary quadratic forms. Moreover, we prove that the sizes of jellyfish are in part dictated by the order of the prime above 2 in certain class groups.