Finite monodromy of some two-parameter families of exponential sums

TL;DR

This paper characterizes when certain exponential sum families over finite fields have finite monodromy, focusing on binomials and Belyi-type polynomials, advancing understanding of their algebraic and geometric properties.

Contribution

It explicitly determines the polynomials for which the associated local systems have finite monodromy, specifically for binomials and Belyi-type polynomials.

Findings

Finite monodromy occurs for specific binomials with particular parameters.

Finite monodromy is characterized for Belyi-type polynomials with certain structures.

Results provide new classifications of exponential sum families with finite monodromy.

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.