Multiplicative character sums and Kloosterman sheaves

Abstract

We are given a prime p, a power q of p, and a prime to p integer a with $q>a\ge 2$. For a nontrivial multiplicative character χ, we consider the one parameter family of character sums$t\mapsto -\sum _x\chi (x^q+x^a-t),$ which are the traces of a local system on the $G_m/F_p\left(\chi \right)$ of nonzero t's. We show that this local system is the pullback of a Kloosterman sheaf $K_{q,a,\rho }$ (any ρ with ${\rho }^{q-a}=\chi$), and determine the geometric monodromy group $G_{\mathrm{geom}}$ of this $K$. We also determine $G_{\mathrm{geom}}$ for the universal family $F_{\chi ,e}$ of sums $-{\sum }_x\chi \left(f_e\right(x\left)\right)$, as $f_e$ runs over degree e polynomials with all distinct roots. These local systems $F_{\chi ,e}$ were the main focus of [14, Chapter 4], and our new results for $F_{\chi ,e}$ are the complete determination of $G_{\mathrm{geom}}$ in the cases where $G_{\mathrm{geom}}$ is finite.