Moduli of abelian varieties

关于交换簇的模的研究项目包括三个部分：正特征p中的Hecke轨道问题，有限域上交换簇的CM提升和p-进拓扑中的Hecke对称性。这里p表示质数。PI以前赞助的研究已经建立了Siegel模变种的Hecke轨道猜想。针对Hecke轨道问题发展的几种方法已被证明对其他问题有用。对特征p和混合特征(0，p)的Hecke对称性的研究，可能会给模簇的几何带来新的工具和新的见解。关于低维亚簇的所有Hecke平移并的预期非密度陈述的进展将为N.Katz提出的关于在不同于任何Jacobian簇的所有代数数域上存在3维或更高维的阿贝尔簇的问题提供肯定的答案。特征为0的正规域上存在提升到同源的CM的一个充要条件已从以前的研究中得知。这一部分的重点是在特征(0，p)的非正规局部区域上存在提升到同源的CM。射影代数簇基本上是由齐次多项式方程组定义的射影空间的子集，周围的射影空间被剥离，只留下抽象的数学结构。阿贝尔簇是一类非常特殊的射影代数簇，可以对其进行加法和减法运算，并且满足标准的算术规则。上面讨论的模空间是一个代数簇，它的点是具有固定维度和对称形状的阿贝尔簇的参量化。这样的模空间没有许多通常类型的对称性，这些对称性与模空间的每个点相关联。取而代之的是，在这些模空间上有许多另一种类型的对称性，称为Hecke对应，它与模空间的每个点相关联。关于Hecke对称性的一个基本问题如下。从模空间的一个点或更一般的一个子簇开始，然后使用所有的Hecke对应将其传播开来。得到的子集是否几乎填满了整个模空间，使得模空间的每个点都有许多点在附近，并尽可能接近它？这个问题主要是在两种情况中的一种：在环上考虑多项式方程，使得素数p乘以1等于0，或者在包含标准整数但可被p的高次幂整除的元素接近0的环上。CM提升问题涉及上述两种环。阿贝尔簇在现代数论中有着广泛的应用，而上述两种环上的模空间的性质往往是数论中应用的关键。PI以前赞助的研究使Hecke对称性成为数论和代数几何的有用工具，给出了已知定理的简短新证明，以及以前方法无法获得的结果。这个项目一旦实施，可能会增加我们对交换变种的模空间几何的了解，并可能产生适用于其他数学领域的其他方法。

This research project on the moduli of abelian varieties contains three components: the Hecke orbit problem in a positive characteristic p, CM lifting of abelian varieties over finite fields and Hecke symmetry in the p-adic topology. Here p denotes a prime number. The PI's previous sponsored research has established the Hecke orbit conjecture for the Siegel modular varieties. Several methods developed for the Hecke orbit problem have proved useful for other question. Research on the Hecke symmetry, both in characteristic p and in mixed characteristics (0,p), will likely lead to new tools and perhaps new insight on the geometry of modular varieties.The effect of Hecke symmetry in the p-adic topology has not been systematically studied before. Progress on the expected non-density statement for the union of all Hecke translates of a lower-dimensional subvariety will provide an affirmative answer to a question posed by N. Katz on the existence of abelian varieties of dimension 3 or higher over the field of all algebraic numbers which are not isogenous to any Jacobian variety. A necessary and sufficient condition for the existence of a CM lifting up to isogeny over a normal domain of characteristic 0 is known from prior sponsored research. The focus in this part of the project is the existence of a CM lifting up to isogeny over a not-necessarily normal local domain of characteristics (0,p).A projective algebraic variety is basically a subset of the projective space defined by a system of homogeneous polynomial equations, with the surrounding projective space stripped off and only the abstract mathematical structure left. An abelian variety is a very special kind of projective algebraic variety such that one can perform addition and substraction on it and the standard rules of arithmetic are satisfied.A moduli space considered above is an algebraic variety whose points parametrizes abelian varieties with a fixed dimension and shape of symmetry. Such a moduli space does not have many symmetries of the usual kind, those which to every point of the moduli space associate another point of the moduli space. Instead there are many symmetries of another sort on these moduli spaces, known as Hecke correspondences, which to every point of the moduli space associates several points of the moduli space.A basic question about Hecke symmetry is the following. Start with either one point or more generally a subvariety of a moduli space, then spread it around using all Hecke correspondences. Does the resulting subset almost fill the whole moduli space, so that every point of the moduli space has many points of the spread-out subset nearby and as close to it as one wants? This question is of interest mainly in one of two situations: the polynomial equations are considered over a ring such that either a prime number p times 1 is equal to 0, or over a ring which contains the standard integers but elements divisible by a high power of p is close to 0.The CM lifting question involves both kinds of rings above. Abelian varieties are extensively used in modern number theory, and properties of the moduli spaces over either kind of rings above are often of key importance for application in number theory. The PI's prior sponsored research has made Hecke symmetry a useful tool for number theory and algebraic geometry, producing short new proofs of known theorems as well as results inaccessible by previous methods. This project, when carried out, is likely to increase our knowledge about geometry of the moduli spaces of abelian varieties and may generate additional methods applicable to other areas of mathematics.