  The purpose of this article is to prove a "Newton over Hodge" result for
finite characters on curves. Let $X$ be a smooth proper curve over a finite
field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an
affine curve. Consider a nontrivial finite character $\rho:\pi_1^{et}(V) \to
\mathbb{C}^\times$. In this article, we prove a lower bound on the Newton
polygon of the $L$-function $L(\rho,s)$. The estimate depends on monodromy
invariants of $\rho$: the Swan conductor and the local exponents. Under certain
nondegeneracy assumptions this lower bound agrees with the irregular Hodge
filtration introduced by Deligne. In particular, our result further
demonstrates Deligne's prediction that the irregular Hodge filtration would
force $p$-adic bounds on $L$-functions. As a corollary, we obtain estimates on
the Newton polygon of a curve with a cyclic action in terms of monodromy
invariants.
