Journal of the Ramanujan Mathematical Society

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Hyperelliptic Curves over 𝔽q and Gaussian Hypergeometric Series


Affiliations
1 Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India
2 Indian Institute of Information Technology, Ambari, Gopinath Bordoloi (G.N.B.) Road, Guwahati-781001, India
     

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Let d≥2 be an integer. Denote by Ed and E'd the hyper-elliptic curves over 𝔽q given by
Ed: y2 = xd + ax + b and E'd: y2 = xd + axd−1 + b,
respectively. We explicitly find the number of 𝔽q-points on Ed and E'd in terms of special values of d Fd-1 and d-1Fd-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+1Fn 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.
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  • Hyperelliptic Curves over 𝔽q and Gaussian Hypergeometric Series

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Authors

Rupam Barman
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India
Gautam Kalita
Indian Institute of Information Technology, Ambari, Gopinath Bordoloi (G.N.B.) Road, Guwahati-781001, India

Abstract


Let d≥2 be an integer. Denote by Ed and E'd the hyper-elliptic curves over 𝔽q given by
Ed: y2 = xd + ax + b and E'd: y2 = xd + axd−1 + b,
respectively. We explicitly find the number of 𝔽q-points on Ed and E'd in terms of special values of d Fd-1 and d-1Fd-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+1Fn 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.