On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2
On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2
复制标题
特征 p > 2 场上超椭圆曲线的雅可比变体
DOI：
10.1016/0021-8693(78)90247-8
复制
发表时间：
1978
影响因子：
0.9
通讯作者：
N. Yui
中科院分区：
文献类型：
--
作者：
N. Yui
众所周知，定义在特征p> 0的域K上的维数为g的阿贝尔簇X产生维数为g且高度为2g的p-可整除群X（p）。设I '是X的群律关于原点局部参数系的幂级数展开而得到的形式群。则I '是X（p）中的连通p-可除群，r具有g和2g之间的任何高度（参见图10）。Tate [14]）。本文主要研究特征p> 2的域上超椭圆曲线C的雅可比簇J（C）。本文的目的是（i）借助于C的Cartier-Manin矩阵A确定p-可分群J（p）和J（C）的形式群r（直到Isn）的结构，（ii）研究J（C）的代数（整体）结构的信息能从形式（局部）结构中恢复多少。在第二节中，我们定义了特征p> 2的理想域上的超椭圆曲线C的Cartier-Manin矩阵A，它遵循Cartier [I]和Manin [S]。然后我们证明了A与C的Hasse-Witt矩阵一致。本文还讨论了A的一些基本但重要的性质。在此之后，在接下来的章节中，我们假设k是一个具有pa（a 3 1）个元素的有限域。在第三节中，我们给出了C的“普通”Jacobi簇J（C）的一个完整刻画。当j（C）是普通的时，C的Cartier-Manin矩阵A完全决定了形式结构J（p），并且在某些情况下也决定了代数结构（直到i = 1）。在本文的其余部分，我们研究了超椭圆曲线C的Jacobi簇J（C），其Cartier-Manin矩阵在R中的行列式为零。在第4节中，我们观察到C的Cartier-Manin矩阵A不再提供足够的信息;它是J（C）相对于k的Frobenius自同态的特征多项式的特征值的p-adic指数，
It is well known that an Abelian variety X of dimension g defined over a field K of characteristic p> 0 yields a p-divisible group X (p) of dimension g and of height 2g. Let I’be the formal group obtained by expansion into power series of the group law of X relative to some system of local parameters at the origin. Then I’is nothing but the connected p-divisible group in X (p) and r has any height between g and 2g (cf. Tate [14]). In the present paper, we confine ourselves to the study of the Jacobian variety J (C) of a hyperelliptic curve C over a field of characteristic p> 2. Our aims here are (i) to determine the structures of the p-divisible group J (p) and of the formal group r of J (C)(up to isogeny) with the help of the Cartier-Manin matrix A of C, and (ii) to investigate how much information about the algebraic (global) structure of J (C)(up to isogeny) can be recovered from the formal (local) structure.We shall give a brief survey of the paper here. In Section 2, we define the Cartier-Manin matrix A of a hyperelliptic curve C over a perfect field of characteristic p> 2 following Cartier [I] and Manin [S]. We then show that A coincides with the Hasse-Witt matrix of C. Some basic but important properties of A are also discussed. After this, throughout the forthcoming sections, we assume that k is a finite field with pa (a 3 1) elements. In Section 3, we give a complete characterization of the “ordinary” Jacobian variety J (C) of C. When j (C) is ordinary, the Cartier-Manin matrix A of C completely determines the formal structure J (p), and in certain cases the algebraic structure as well (up to isogeny). In the rest of the paper, we study the Jacobian variety J (C) of the hyperelliptic curve C whose Cartier-Manin matrix has determinant zero in R. In Section 4, we observe that the Cartier-Manin matrix A of C no longer provides enough information; it is the p-adic exponents of the eigenvalues of the characteristic polynomial of the Frobenius endomorphism of J (C) relative to k that determine