Hypergeometric functions and a family of algebraic curves

TL;DR

This paper explores the connections between hypergeometric functions and algebraic curves defined by specific equations, relating periods, point counts over finite fields, and providing new proofs for existing results.

Contribution

It introduces a new framework linking hypergeometric series with algebraic curves and their point counts, extending previous work with novel proofs.

Findings

Relation between periods of algebraic curves and hypergeometric series

Expression of point counts over finite fields using Gaussian hypergeometric series

Alternative proof of a known result in the theory of algebraic curves

Abstract

Let $\lambda \in \mathbb{Q}\setminus \{0, 1\}$ and $l \geq 2$, and denote by $C_{l,\lambda}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=x(x-1)(x-\lambda).$$ In this paper we define $\Omega(C_{l, \lambda})$ analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on $C_{l, \lambda}$ over a finite field and Gaussian hypergeometric series. Finally we give an alternate proof of a result of \cite{rouse}.