Let λ ∈ Q \ {0, −1} and l ≥ 2. Denote by C l,λ the nonsingular projective algebraic curve over Q with affine equation given byIn this paper we give a relation between the number of points on C l,λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [10]. We find some special values of 3 F 2 and 2 F 1 Gaussian hypergeometric series. Finally we evaluate the value of 3 F 2 (4) which extends a result of [11].

Let λ ∈ Q \ {0, 1} and l ≥ 2, and denote by C l,λ the nonsingular projective algebraic curve over Q with affine equation given byIn this paper we define Ω(C l,λ ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on C l,λ over a finite field and Gaussian hypergeometric series. Finally we give an alternate proof of a result of [13].

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We present explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of 2 F 1hypergeometric functions over F q for q ≡ 1(mod 6) and q ≡ 1(mod 4).

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