On the number of zeros of diagonal cubic forms over finite fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y,z\in {\mathbb{F}}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^3+\cdots +x_s^3=z$ and $x_1^3+\cdots +x_{s-1}^3+y{x}_s^3=0$, respectively. Gauss proved that if $q=p,p\equiv 1(\mathrm{mod}3)$ and $y$ is non-cubic, then $T_3(y)=p^2+\frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,c\equiv 1(\mathrm{mod}3)$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${\mathbb{F}}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${\mathbb{F}}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${\mathbb{F}}_p$. Furthermore, we show that the generating functions ${\sum }_{s=1}^{\infty }{N}_s(z)x^s$ and ${\sum }_{s=1}^{\infty }{T}_s(y)x^s$ are rational functions for any $z,y\in {\mathbb{F}}_q^{\ast }:={\mathbb{F}}_q\setminus \{0\}$ with $y$ being non-cubic over ${\mathbb{F}}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.

13 pages