On Cubic Exponential Sums and Gauss Sums

Let e_q be a nontrivial additive character of a finite field 𝔽_q of order q ≡ 1(mod 3) and let ψ be a cubic multiplicative character of 𝔽_q, ψ(0) = 0. Consider the cubic Gauss sum and the cubic exponential sum

$$ G\left(\psi \right)=\sum \limits_{z\in {\mathbb{F}}_q}{e}_q(z)\psi (z),\kern0.5em C\left(\omega \right)=\sum \limits_{z\in {\mathbb{F}}_q}{e}_q\left(\frac{z^3}{\omega }-3z\right),\kern0.5em \omega \in {\mathbb{F}}_q,\kern1em \omega \ne 0. $$

It is proved that for all nonzero a, b ∈ 𝔽_q,

$$ \frac{1}{q}\sum \limits_nC(an)C(bn)\psi (n)+\frac{1}{q}\psi (ab)G{\left(\psi \right)}^2=\overline{\psi}(ab)\psi \left(a-b\right)\overline{G\left(\psi \right)}, $$

where the summation runs over all nonzero n ∈ 𝔽_q.