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

Abstract Let 𝔽q{\mathbb{F}_{q}} be the finite field of q=pk{q=p^{k}} elements with p being a prime and let k be a positive integer. For any y,z∈𝔽q{y,z\in\mathbb{F}_{q}}, let Ns⁢(z){N_{s}(z)} and Ts⁢(y){T_{s}(y)} denote the numbers of zeros of x13+⋯+xs3=z{x_{1}^{3}+\cdots+x_{s}^{3}=z} and x13+⋯+xs-13+y⁢xs3=0{x_{1}^{3}+\cdots+x_{s-1}^{3}+yx_{s}^{3}=0}, respectively. Gauss proved that if q=p{q=p}, p≡1(mod3){p\equiv 1~{}(\bmod~{}3)} and y is non-cubic, then T3⁢(y)=p2+12⁢(p-1)⁢(-c+9⁢d),T_{3}(y)=p^{2}+\frac{1}{2}(p-1)(-c+9d), where c and d are uniquely determined by 4⁢p=c2+27⁢d2{4p=c^{2}+27d^{2}} and c≡1(mod3){c\equiv 1~{}(\bmod~{}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 𝔽p{\mathbb{F}_{p}}. But the sign problem is kept open for the remaining case of 2 being cubic in 𝔽p{\mathbb{F}_{p}}. In this paper, we solve this sign problem by determining the sign of d when 2 is cubic in 𝔽p{\mathbb{F}_{p}}. Furthermore, we show that the generating functions ∑s=1∞Ns⁢(z)⁢xs{\sum_{s=1}^{\infty}N_{s}(z)x^{s}} and ∑s=1∞Ts⁢(y)⁢xs{\sum_{s=1}^{\infty}T_{s}(y)x^{s}} are rational functions for any z,y∈𝔽q*:=𝔽q∖{0}{z,y\in\mathbb{F}_{q}^{*}:=\mathbb{F}_{q}\setminus\{0\}} with y being non-cubic over 𝔽q{\mathbb{F}_{q}}, and we also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.

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