On Families of Pure Slope $L$-Functions

Abstract

Let $R$ be the ring of integers in a finite extension $K$ of ${\mathbb{Q}}_p$, let $k$ be its residue field and let $\chi :{\pi }_1(X)\to R^{\times }=G{L}_1(R)$ be a "geometric" rank one representation of the arithmetic fundamental group of a smooth affine $k$-scheme $X$. We show that the locally $K$-analytic characters $\kappa :R^{\times }\to {\mathbb{C}}_p^{\times }$ are the ${\mathbb{C}}_p$-valued points of a $K$-rigid space $\mathcal{W}$ and that $$L(\kappa \circ \chi ,T)=\prod _{\overline{x}\in X}\frac{1}{1-(\kappa \circ \chi )(Fro{b}_{\overline{x}})T^{°(\overline{x})}},$$viewed as a two variable function in $T$ and $\kappa$, is meromorphic on ${\mathbb{A}}_{{\mathbb{C}}_p}^1\times \mathcal{W}$. On the way we prove, based on a construction of Wan, a slope decomposition for ordinary overconvergent (finite rank) $\sigma$-modules, in the Grothendieck group of nuclear $\sigma$-modules.

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• Lectures on zeta functions over finite fields