P-Adic estimates of abelian ArtinL-functions on curves

Abstract

AbstractThe purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. LetXbe 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 theL-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 forcep-adic bounds onL-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.