p-adic超几何函数与Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟,与许多数论问题都有联系.设Fq是q元有限域,λ∈Fq,n为正整数.本文研究了Dwork超曲面Dλ^n:x1^n+x2^n+…+xn^n=nλx1x2…xn及其推广形式上的Fq-有理点,并在n与q(q-1)互素时给出了由p-adic超几何函数表示的各种Fq-有理点个数的公式,从而修正和改进了Barman与Goodson等人的结论.

p-adic Hypergeometric Functions and Rational Points on Dwork Hypersurfaces

p-adic hypergeometric functions are hypergeometric functions over finite fields analogous to the classical Gaussian hypergeometric functions,which have been found applications in diverse number theory problems.Let Fq be the finite field of q elements,λ∈Fq and n be a positive integer.This paper investigates the Fq-rational points on the Dwork hypersurface Xλ^n:x1^n+x2^n+…+xnn=nλx1x2…xn as well as its generalized form,and provides the formula for the number of the Fq-rational points in terms of a p-adic hypergeometric function when n and q(q-1)are coprime,which revises and improves the results given by Barman and Goodson et al.