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 \(\rho :\pi _1(X) \rightarrow \mathbb {C}_p^\times \) be a finite character of order \(p^n\). By previous work of the first author, the Newton polygon \({{\,\mathrm{\text {NP}}\,}}(\rho )\) lies above a ‘Hodge polygon’ \({{\,\mathrm{\text {HP}}\,}}(\rho )\) defined using ramification invariants of \(\rho \). In this article we study the contact between these two polygons. We prove that \({{\,\mathrm{\text {NP}}\,}}(\rho )\) and \({{\,\mathrm{\text {HP}}\,}}(\rho )\) 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 \({{\,\mathrm{\text {NP}}\,}}(\rho )\) and \({{\,\mathrm{\text {HP}}\,}}(\rho )\).