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arXiv · 1208.0492

Hypergeometric functions and a family of algebraic curves

Rupam Barman·Gautam Kalita
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Abstract

Let $λ\in \mathbb{Q}\setminus \{0, 1\}$ and $l \geq 2$, and denote by $C_{l,λ}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=x(x-1)(x-λ).$$ In this paper we define $Ω(C_{l, λ})$ 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 $C_{l, λ}$ over a finite field and Gaussian hypergeometric series. Finally we give an alternate proof of a result of \cite{rouse}.

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Rupam Barman, Gautam Kalita. 2012-08-02. Hypergeometric functions and a family of algebraic curves. https://arxiv.org/abs/1208.0492

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