Abstract: Let $C$ be a curve over a finite field and let $\rho$ be a
nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the
Artin $L$-function associated to $\rho$ is a polynomial with algebraic
coefficients. Furthermore, the roots of this polynomial are
$\ell$-adic units for $\ell \neq p$ and have Archemedian absolute
value $\sqrt{q}$. Much less is known about the $p$-adic properties of
these roots, except in the case where the image of $\rho$ has order
$p$. We prove a lower bound on the $p$-adic Newton polygon of the
Artin $L$-function for any representation in terms of local monodromy
decompositions. If time permits, we will discuss how this result
suggests the existence of a category of wild Hodge modules on Riemann
surfaces, whose cohomology is naturally endowed with an irregular
Hodge filtration.