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 $\chi:\pi_1(X)\to R^{\times}=GL_{1}(R)$ 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 ${\cal W}$ and that $$L(\kappa\circ\chi,T)=\prod_{\overline{x}\in X}\frac{1}{1-(\kappa \circ\chi)(Frob_{\overline{x}})T^{\deg(\overline{x})}},$$viewed as a two variable function in $T$ and $\kappa$, is meromorphic on $\mathbb{A}_{\mathbb{C}_p}^1\times{\cal W}$. 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.