Elliptic curves and special values of Gaussian hypergeometric series

DOI10.1016/J.JNT.2013.03.010zbMATH Open1295.11137arXiv1311.4695OpenAlexW1969129728MaRDI QIDQ740854

Author name not available (Why is that?)

Publication date: 9 September 2014

Published in: (Search for Journal in Brave)

Abstract: Let

d g e q 2

be an integer. Denote by

E_{d}

and

E'_{d}

the hyperelliptic curves over

m a t h b b F_{q}

given by

E_d: y^2=x^d+ax+b~~~ ext{and} ~~~E'_d: y^2=x^d+ax^{d-1}+b,

respectively. We explicitly find the number of

m a t h b b F_{q}

-points on

E_{d}

and

E'_{d}

in terms of special values of

_{d} F_{d - 1}

and

_{d - 1} F_{d - 2}

Gaussian hypergeometric series with characters of orders

d - 1

,

d

,

2 (d - 1)

,

2 d

, and

2 d (d - 1)

as parameters. This gives a solution to a problem posed by Ken Ono cite[p. 204]{ono2} on special values of

_{n + 1} F_{n}

Gaussian hypergeometric series for

n > 2

. We also show that the results of Lennon cite{lennon1} and the authors cite{BK3} on trace of Frobenius of elliptic curves follow from the main results.

Full work available at URL: https://arxiv.org/abs/1311.4695

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