Construction of Curves with a Controlled First Slope using p-Symmetric Numbers
Robert Moore, Hui June Zhu
TL;DR
This work links the first Newton slope of Artin-Schreier curves $X_f: y^p - y = f(x)$ to the $p$-adic weight of the support of $f$, establishing that the lower bound $1/s_p(\nu)$ is attained exactly when the unique maximal weight element $\nu$ in $\mathrm{Supp}(f)$ is $p$-symmetric. The authors develop a $p$-adic change-making framework within Dwork theory, defining minimizers and their heights to derive sharp valuations for the characteristic series of the Dwork operator and, consequently, the Newton slopes of $L(f,s)$ and $Z(X_f/\mathbb{F}_q,s)$. A key contribution is a criterion: if $\nu$ is $p$-symmetric, the first slope equals $1/s_p(\nu)$ with multiplicity controlled by the minimizer height; if $\nu=p^k-1$, the first slope is $1/(k(p-1))$ with multiplicity $k(p-1)$. Leveraging this, the paper constructs explicit families of curves with first slope $1/n$ for every $n\ge 2$ in any characteristic $p$, enriching the landscape of non-supersingular Newton polygons and connecting combinatorial $p$-adic properties to geometric invariants.
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.