Hyperelliptic curves over \mathbbF_q and Gaussian hypergeometric series

2013/11/19 by Barman, Rupam, Kalita, Gautam
#11G20 #11T24 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1311.4695

Abstract

Let d≥2 be an integer. Denote by E_d and E'_d the hyperelliptic curves over \mathbbF_q given by E_d: y²=x^d+ax+b~~~ and ~~~E'_d: y²=x^d+ax^d-1+b, respectively. We explicitly find the number of \mathbbF_q-points on E_d and E'_d in terms of special values of _dF_d-1 and _d-1F_d-2 Gaussian hypergeometric series with characters of orders d-1, d, 2(d-1), 2d, and 2d(d-1) as parameters. This gives a solution to a problem posed by Ken Ono \cite[p. 204]ono2 on special values of _n+1F_n Gaussian hypergeometric series for n > 2. We also show that the results of Lennon \citelennon1 and the authors \citeBK3 on trace of Frobenius of elliptic curves follow from the main results.

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