Hecke orbits as Shimura varieties in positive characteristic
Hecke orbits as Shimura varieties in positive characteristic
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赫克轨道作为志村品种的积极特征
DOI：
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发表时间：
2006
期刊：
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通讯作者：
C. Chai
中科院分区：
文献类型：
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作者：
C. Chai
设p是素数，M是Fp上的PEL型模簇
其将具有固定PEL型额外对称性的特征p中的阿贝尔变种分类。
考虑具有额外对称性的p-可分群，该群由所有p次方挠率组成
M上的泛阿贝尔格式M中的轨迹对应于固定同构
F·奥尔特称具有额外对称性的p-可分群的类型为叶。每片叶子都是一片
模簇M的光滑局部闭子簇，它在所有素数到p下都是稳定的
关于M.Oort的Hecke对应猜想每个Hecke轨道在叶中是稠密的
控制住它。为这一猜想而设计的工具包括(A)刚性，(B)全局单行，
和(C)正则坐标。正则坐标理论推广了经典的
Serre�泰特坐标；它断言，在局部喷气空间的水平上，每一片树叶都是
以规范的方式从p-可分形式群到有限的纤维族。这个
当M是主要分类的Siegel模簇时，证明了Hecke轨道猜想
固定维极化阿贝尔簇，当M是Hilbert模簇时
用实数乘法对阿贝尔品种进行分类。Siegel案的证明，与
F.Oort，利用Hilbert模变种中非超奇异叶的不可约性，
C.-F.Yu。这一证明在很大程度上依赖于Siegel模簇的一个特殊性质：
Siegel模簇Ag，n的FP-有理点被Hilbert的FP-有理点填满
包含在Ag，n中的模数变种。进一步发展的可能方向包括直线型
亚变种和p一单倍体。这篇文章的标题表明，每一片叶子都值得
被视为具有p特色的下村变种。
Let p be a prime number, and let M be a modular variety of PEL type over Fp
which classifies abelian varieties in characteristic p with extra symmetries of a fixed PEL type.
Consider the p-divisible group with extra symmetries consisting of all p-power torsions of
the universal abelian scheme over M. The locus in M corresponding to a fixed isomorphism
type of a p-divisible group with extra symmetry is called a leaf by F. Oort. Each leaf is a
smooth locally closed subvariety of the modular varietyM which is stable under all prime-to-p
Hecke correspondences on M. Oort conjectured that every Hecke orbit is dense in the leaf
containing it. Tools fashioned for this conjecture include (a) rigidity, (b) global monodromy,
and (c) canonical coordinates. The theory of canonical coordinates generalizes the classical
Serre�Tate coordinates; it asserts that locally at the level of jet-spaces, every leaf is built up
from p-divisible formal groups through a finite family of fibrations in a canonical way. The
Hecke orbit conjecture is affirmed when M is a Siegel modular variety classifying principally
polarized abelian varieties of a fixed dimension, and also when M is a Hilbert modular variety
classifying abelian varieties with real multiplications. The proof of the Siegel case, joint with
F. Oort, uses the irreducibility of non-supersingular leaves in Hilbert modular varieties due to
C.-F.Yu. That proof relies heavily on a special property of Siegel modular varieties: The set of
Fp-rational points of a Siegel modular variety Ag,n is filled up by Fp-rational points of Hilbert
modular varieties contained in Ag,n. Possible directions for further progress include Tate-linear
subvarieties and p-adic monodromy. The title of this article suggests that each leaf deserves to
be viewed as a Shimura variety in characteristic p in its own right.