2023/02/21 by Eleanor McSpirit, Ken Ono, McSpirit, Eleanor +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2302.10387
openalex publication_date 2023/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
One of the most celebrated applications of Gauss' 2F1 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 \citejelly1 in which a finite field analogue of AGM_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.