ON THE POLYNOMIAL x^d + ax + b OVER 𝔽_q AND GAUSSIAN HYPERGEOMETRIC SERIES

Abstract

For d ≥ 2, denote by P d (x) the polynomial over Fq given byWe explicitly find the number of solutions in Fq of the polynomial equation P d (x) = 0 in terms of special values of d F d−1 and d−1 F d−2 Gaussian hypergeometric series with characters of orders d and d − 1 as parameters. This solves a problem posed by K. Ono (see p. 204 in [Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, No. 102 (American Mathematical Society, Providence, RI, 2004)]) on special values of n+1 Fn Gaussian hypergeometric series for n > 2.

Hypergeometric functions and algebraic curves $y^e=x^d+ax+b$

“…It is found that these functions satisfy many summation and transformation formulas analogous to classical hypergeometric series. In a series of papers, many interesting relations have been established between special values of these hypergeometric functions and the number of points on certain algebraic curves over finite fields (see, for examples, [1,2,3,4,5,6,10,9,12,14]).…”

“…Fuselier [6] and Lennon [12] found formulas for the trace of Frobenius endomorphism of a certain family of elliptic curves in terms of Gaussian hypergeometric series containing characters of order 12. In [4], Barman and Kalita found the number of solutions of the polynomial equation x d + ax + b = 0 over a finite field F q in terms of special values of Gaussian hypergeometric series with characters of orders d and d − 1 under the condition that q ≡ 1(mod d(d − 1)) and d ≥ 2. The same authors, in [5], expressed the number of F q -points on a hyperelliptic curve in terms of special values of Gaussian hypergeometric series.…”