Construction of curves with a controlled first slope using p-symmetric numbers

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