Fermat quotients, correspondences, and uniformization

费马商、对应和均匀化

• 批准号: 0552314
• 负责人: Alexandru Buium
• 金额: $ 11.21万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0552314&HistoricalAwards=false
• 关键词: Fermat; quotients; correspondences; uniformization

Given a geometric object and an equivalence relation on it one is often faced with the situation that there are no invariant functions except the constants. In particular the categorical quotient reduces to a point. One way to go around this difficulty is to shift focus from the study of invariant functions to the study of the ``non-effective descent data'' for the equivalence relation. Stack theory and non-commutative geometry are both examples of this strategy. In previous work, the PI proposed a rather different approach. The idea is to enlarge the repertoire of functions of algebraic geometry by adjoining a ``Fermat quotient operator''; the new functions in the repertoire(which should be viewed as arithmetic analogues of non-linear differential operators) turn out to be sufficiently flexible to sometimes provide interesting invariants. The new resulting geometry can be referred to as arithmetic differential geometry. This is a commutative geometry and can be viewed as an arithmetic analogue of the Ritt-Kolchin differential algebraic geometry. There is an arithmetic differential geometry for each prime number. The main conjecture proposed by the PI is that if one is given an algebraic curve over a number field and a correspondence on it satisfying a certain natural density condition then the categorical quotient of the curve by the correspondence is non-trivial in arithmetic differential geometry for almost all primes if, and only if, over the complex numbers, the correspondence admits a ``complex analytic uniformization''. One can complement the conjecture above by predicting that in most cases the corresponding quotient spaces are ``rational''. Previous work of the PI led to confirmation of the conjecture in a number of basic examples. The present research project proposes to continue the work on this conjecture, to extend this work to the higher dimensional case and to the ``partial differential case", and to exploit the interactions of this theory with other theories. The construction of quotient spaces is a central problem in geometry. There are natural situations when quotients, in a given geometry, exhibit fundamental pathologies. This prompts one toseek extensions of classical geometries where the quotient pathologies can be avoided. The proposal puts forward a new extension of algebraic geometry which seems well suited to treatpathologies arising from arithmetical problems.

给定一个几何对象及其上的等价关系，人们常常会遇到除了常量之外没有不变函数的情况。特别地，范畴商减少到一个点。解决这一困难的一个方法是将重点从研究不变函数转移到研究等价关系的“非有效下降数据”上。堆栈理论和非交换几何都是这种策略的例子。在之前的工作中，PI提出了一种完全不同的方法。其思想是通过引入“费马商算子”来扩充代数几何的函数库；保留表中的新函数（应被视为非线性微分运算符的算术类似物）具有足够的灵活性，有时可以提供有趣的不变量。所得到的新几何可以称为算术微分几何。这是一个交换几何，可以看作是Ritt-Kolchin微分代数几何的算术类比。每个素数都有一个等差微分几何。PI所提出的主要猜想是，如果给定一个数域上的代数曲线及其上的对应关系满足一定的自然密度条件，那么对于几乎所有素数，该对应关系的范畴商在算术微分几何中是非平凡的，当且仅当在复数上，该对应关系允许“复解析一致化”。我们可以通过预测在大多数情况下相应的商空间是“有理”来补充上述猜想。PI先前的工作在一些基本的例子中证实了这个猜想。本课题拟继续对这一猜想进行研究，将这一工作扩展到高维情况和偏微分情况，并探索这一理论与其他理论的相互作用。商空间的构造是几何中的一个中心问题。在给定的几何中，商表现出基本的病态是很自然的。这促使人们寻求经典几何的扩展，从而避免商病态。该建议提出了代数几何的一个新的扩展，似乎很适合治疗由算术问题引起的病理。