Acta Mathematica Vietnamica

We determine in many (but not all) cases the geometric monodromy group $G_\textrm{geom}$ of the one-parameter local system $\mathcal {G}(f,a,\chi )$ on $\mathbb {G}_m$ in characteristic $p >0$, whose trace function (at $t \in L^\times$ for large enough finite extensions L of the field $\mathbb {F}_p$) is $$\begin{aligned} t \in L^\times \mapsto -\sum _{x \in L^\times }\psi (f(x)+t/x^a)\chi (x), \end{aligned}$$ with $\psi$ a (nontrivial) additive character, f(x) a polynomial of degree $A \ge 1$, $a \ge 1$, and $\chi$ a (possibly trivial) multiplicative character. These local systems are generalizations of Kloosterman sheaves.

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