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arXiv · 2411.01832

Construction of Curves with a Controlled First Slope using p-Symmetric Numbers

Robert Moore·Hui June Zhu
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Abstract

This paper establishes a constructive link between the first slope of Artin-Schreier curves X_f: y^p-y=f(x) and the p-adic weight of the support of f(x). If the maximal p-adic weight element v in Supp(f) is unique, we show that the first slope's lower bound of 1/s_p(v) is achieved if and only if v satisfies a combinatorial p-adic condition, which we define as p-symmetry. As an application, we construct explicit families of curves in every characteristic p with first slope equal to 1/n for every n>2.

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Robert Moore, Hui June Zhu. 2024-11-04. Construction of Curves with a Controlled First Slope using p-Symmetric Numbers. https://arxiv.org/abs/2411.01832

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