Hypergeometric functions and a family of algebraic curves
For \(\lambda\in\mathbb Q\backslash\{0,1\}\) and \(l\geq 2\), let \(C_{l,\lambda}\) be the nonsingular projective curve over \(\mathbb Q\) with affine equation \[ y^l=x(x-1)(x-\lambda). \] The paper in review defines a quantity \(\Omega(C_{l,\lambda})\) that generalizes the real periods of elliptic curves (the case for elliptic curves is \(l=2\)), and gives relations between \(\Omega(C_{l,\lambda})\), the number of points on \(C_{l,\lambda}\), and the Gaussian hypergeometric series. These relations generalize those for the elliptic curves in [\textit{K. Ono}, Trans. Am. Math. Soc. 350, No. 3, 1205--1223 (1998; Zbl 0910.11054)]. Finally, the paper gives an alternative proof of a result of \textit{J. Rouse} [Ramanujan J. 12, No. 2, 197--205 (2006; Zbl 1193.11057)].
scientific article; zbMATH DE number 6088841
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