Previous issue ·  Next issue ·  Recently posted articles ·  Most recent issue · All issues   
Home Overview Authors Editorial Contact Subscribe

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