具有常秩Cartier-Manin矩阵的Artin-Schreier曲线族
Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank
摘要：
设k是特征p > 0的代数闭域。对于k（x）中的某个f（x），每个Artin-Schreier k-曲线X都有一个形式为y^p - y = f（x）的方程，使得p不能整除f（x）极点阶数的最小公倍数L。在p与1 mod L全等的条件下，朱文龙证明了X的L-函数的牛顿多边形由f（x）的Hodge多边形决定。特别地，牛顿多边形仅取决于f（x）的极点的阶数，而不取决于极点的位置或f（x）的系数。本文证明了关于X的Jacobian的p-挠群方案的a-数的一个类似结果，给出了具有常数a-数的Jacobian族的第一个非平凡例子。等价地，我们考虑X的正则1-形式层上的半线性Cartier算子，并给出了曲线族的第一个非平凡例子，其Cartier-Manin矩阵具有常秩。
英文摘要：
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.
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