Rationality of trace and norm $L$-functions

Abstract

For a given $\ell$-adic sheaf $\mathcal {F}$ on a commutative algebraic group $G$ over a finite field $k$ and an integer $r\geq1$, we define the $r$th local norm $L$-function of $\mathcal {F}$ at a point $t\in G(k)$ and prove its rationality. This function gives information on the sum of the local Frobenius traces of $\mathcal {F}$ over the points of $G(k_{r})$ (where $k_{r}$ is the extension of degree $r$ of $k$) with norm $t$. For $G$ the $1$-dimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties.