Finite monodromy of some families of exponential sums

TL;DR

This paper provides a numerical criterion to determine when certain exponential sums over finite fields have finite monodromy, with explicit examples computed for specific cases.

Contribution

It introduces a new numerical criterion for finite monodromy of $ ext{l}$-adic sheaves associated with exponential sums and applies it to explicit cases.

Findings

Numerical criterion for finite monodromy established

Explicit computations for specific exponential sums

Criteria applicable to a range of cases

Abstract

Given a prime $p$ and an integer $d>1$, we give a numerical criterion to decide whether the $\ell$-adic sheaf associated to the one-parameter exponential sums $t\mapsto \sum_x\psi(x^d+tx)$ over ${\mathbb F}_p$ has finite monodromy or not, and work out some explicit cases where this is computable.