Journal of the Ramanujan Mathematical Society

Rupam Barman ¹, Gautam Kalita ²

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

Abstract

Let d≥2 be an integer. Denote by E_d and E'_d the hyper-elliptic curves over 𝔽_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 𝔽_q-points on E_d and E'_d in terms of special values of _d F_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 [17, p. 204] on special values of _n+1F_n 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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Rupam Barman ¹

Affiliations
1 Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, IN

Abstract

For a fixed topological generator u of 1+pℤ_p, the map f_n:ℤ_p→ℤ_p defined by f_n(x)=(u^x_n) is a continuous function. In this paper we study p-adic properties of certain Mahler coefficients of f_n for different values of n exploiting some combinatoridentities.

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Anupam Saikia ¹, Rupam Barman ²

Affiliations
1 Department of Mathematics, Indian Institute of Technology, Guwahati-781039, Assam, IN
2 Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, IN

Abstract

In this paper we study a relation between the λ-invariants of a p-adic measure and its Γ-transform exploiting certain combinatorial identities. Along the way we also determine p-adic properties of certain Mahler coefficients.

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