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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 \(\rho \).