Hypergeometric functions and a family of algebraic curves

Abstract

Let λ∈ℚ∖{0,1} and l≥2, and denote by Cl,λ the nonsingular projective algebraic curve over ℚ with affine equation given by $$y^l=x(x-1)(x-\lambda).$$ In this paper, we define Ω(Cl,λ) 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 Cl,λ over a finite field and Gaussian hypergeometric series. Finally, we give an alternate proof of a result of Rouse (Ramanujan J. 12(2):197–205, 2006).