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arXiv · 1507.00914

Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves

Mohammad Sadek
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Abstract

We study the character sums \[\phi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } \psi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left((x^{m}+a)(x^{n}+b)\right)\] where $\phi$ is the quadratic character defined over $\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties.

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Mohammad Sadek. 2015-07-03. Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves. https://arxiv.org/abs/1507.00914

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