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.