Newton Polygons of Sums on Curves I: Local-to-Global Theorems

Abstract

The purpose of this article is to study Newton polygons of certain abelian $L$-functions on curves. Let $X$ be a smooth affine curve over a finite field ${\mathbb{F}}_q$ and let $\rho :{\pi }_1(X)\to {\mathbb{C}}_p^{\times }$ be a finite character of order $p^n$. By previous work of the first author, the Newton polygon $\mathrm{NP}(\rho )$ lies above a `Hodge polygon' $\mathrm{HP}(\rho )$, which is defined using local ramification invariants of $\rho$. In this article we study the touching between these two polygons. We prove that $\mathrm{NP}(\rho )$ and $\mathrm{HP}(\rho )$ share a vertex if and only if a corresponding vertex is shared between the Newton and Hodge polygons of `local' $L$-functions associated to each ramified point of $\rho$. As a consequence, we determine a necessary and sufficient condition for the coincidence of $\mathrm{NP}(\rho )$ and $\mathrm{HP}(\rho )$.

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