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

TL;DR

This paper investigates families of Artin-Schreier curves over algebraically closed fields of characteristic p, demonstrating that the Cartier-Manin matrix's rank remains constant within these families, extending known results about Newton polygons and p-torsion group schemes.

Contribution

It establishes the first non-trivial examples of families of Jacobians with constant a-number by analyzing the Cartier operator's rank on Artin-Schreier curves.

Findings

Cartier-Manin matrix rank remains constant in certain families

Provides explicit examples of Jacobians with fixed a-number

Extends understanding of p-torsion group schemes in algebraic geometry

Abstract

Let k be an algebraically closed field of characteristic p > 0. Every Artin-Schreier k-curve X has an equation of the form y^p - y = f(x) for some f(x) in 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 is congruent to 1 mod L, 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.