Moduli of abelian varieties

本研究项目包含三个组成部分：正特征p中的Hecke轨道问题、有限域上阿贝尔变体的CM提升问题和p进拓扑中的Hecke对称性问题。这里p表示质数。PI之前赞助的研究已经为西格尔模变体建立了赫克轨道猜想。为赫克轨道问题开发的几种方法已证明对其他问题也有用。对Hecke对称性的研究，包括特征p和混合特征（0,p），可能会产生新的工具，也许会对模变体的几何有新的认识。Hecke对称性对p进拓扑结构的影响尚未得到系统的研究。关于低维子变体的所有Hecke平移的并的期望非密度陈述的进展，将为N. Katz关于在所有不同于任何雅可比变体的代数数域上存在3维或更高维的阿贝尔变体的问题提供一个肯定的答案。在特征为0的正规域上，CM上升到等基因存在的充分必要条件已经从先前赞助的研究中得到。这部分项目的重点是在特征（0,p）的不一定正规的局部域上存在一个上升到等源的CM。射影代数变体基本上是由齐次多项式方程组定义的射影空间的一个子集，去掉周围的射影空间，只留下抽象的数学结构。阿贝尔变量是一种非常特殊的投影代数变量，可以对其进行加法和减法运算，并且满足标准的算术规则。上面所考虑的模空间是一个代数变体，它的点参数化了具有固定维数和对称形状的阿贝尔变体。这样的模空间没有很多通常的对称性，那些与模空间的每一个点相关联的模空间的另一个点。相反，在这些模空间上有许多另一种对称，称为赫克对应，模空间的每个点都与模空间的几个点相关联。关于赫克对称的一个基本问题如下。从一个点或更一般的模空间的子变种开始，然后使用所有的Hecke对应将其展开。所得到的子集是否几乎填满了整个模空间，以至于模空间的每一个点都有许多展开子集的点在它附近或者在它附近？这个问题主要在以下两种情况中的一种情况下引起人们的兴趣：在一个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。阿贝尔变在现代数论中得到了广泛的应用，而上述两种环上模空间的性质对于数论的应用具有重要的意义。PI先前资助的研究使赫克对称成为数论和代数几何的有用工具，为已知定理提供了简短的新证明，以及用以前的方法无法获得的结果。这个项目，一旦实施，可能会增加我们对阿贝尔变体的模空间的几何知识，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.