# Journal of the Ramanujan Mathematical Society

Volume 30, Issue 3, September 2015 pp. 331–348.

Hyperelliptic curves over Fq and Gaussian hypergeometric series

**Authors**:
Rupam Barman and Gautam Kalita

**Author institution:**Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India

**Summary: **
Let d ≥ 2 be an integer. Denote by Ed and E'd the
hyperelliptic curves over Fq given by
Ed : y2 = xd + ax + b and E'd: y2 = xd + ax d - 1 + b,
respectively. We explicitly find the number of
Fq-points on Ed and E'd in terms of special
values of dFd - 1 and d - 1 F 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 [17, p. 204] on special
values of n + 1 Fn Gaussian hypergeometric series for n >
2. We also show that the results of Lennon [14] and the
authors [4] on trace of Frobenius of elliptic curves follow
from the main
results.

Contents
Full-Text PDF