Finite monodromy of some two-parameter families of exponential sums

Abstract

We determine the set of polynomials $f(x)\in k[x]$, where $k$ is a finite field, such that the local system on ${\mathbb{G}}_m^2$ which parametrizes the family of exponential sums $(s,t)\mapsto {\sum }_{x\in k}\psi (sf(x)+tx)$ has finite monodromy, in two cases: when $f(x)=x^d+\lambda x^e$ is a binomial and when $f(x)=(x-\alpha {)}^d(x-\beta {)}^e$ is of Belyi type.