Families of Artin–Schreier curves with Cartier–Manin matrix of constant rank

Abstract

Let k be an algebraically closed field of characteristic p>0. Every Artin–Schreier k-curve X has an equation of the form yp−y=f(x) for some f(x)∈k(x) such that p does not divide the least common multiple L of the orders of the poles of f(x). Under the condition that p≡1modL, Zhu proved that the Newton polygon of the L-function of X is determined by the Hodge polygon of f(x). In particular, the Newton polygon depends only on the orders of the poles of f(x) and not on the location of the poles or otherwise on the coefficients of f(x). In this paper, we prove an analogous result about the a-number of the p-torsion group scheme of the Jacobian of X, providing the first non-trivial examples of families of Jacobians with constant a-number. Equivalently, we consider the semi-linear Cartier operator on the sheaf of regular 1-forms of X and provide the first non-trivial examples of families of curves whose Cartier–Manin matrix has constant rank.