In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface $$X=\{(x_1,\ldots, x_5)\in \mathbb{P}^4: \sum_ix_i=\sum_ix_i^{-1}=0\}.$$ This $\zeta$-function is somehow related to computing the fifth moment of Kloosterman sums $$\mathcal{S}_5(p)=\sum_{t\in \mathbb{F}_p} \big[K(t)\big]^5,$$ where $$K(t)=\sum_{u\in (\mathbb{F}_p)^{\times}}\mbox{exp}\Big( 2\pi i \,\frac{tu+u^{-1}}{p}\Big).$$
1. I was hoping somebody would illuminate for me the precise relationship between the $\zeta$-function for $X$ and evaluating the number $\mathcal{S}_5(p)$. I imagine that $S_5(p)$ can be somehow reduced to counting the number of points on $X$, but how exactly?
2. Let $a,b,c\in (\mathbb{F}_p)^{\times}$. I was curious about relating the number $$\sum_{t\in \mathbb{F}_p}K(t)K(at)K(bt)K(ct)$$ to the $\zeta$ function of some surface as well. Note that $a=b=c=1$, then this sum is the fourth moment.
Thanks in advance.