Fermat quotients, correspondences, and uniformization

给定一个几何对象和它的等价关系，人们经常面临这样的情况，即除了常数之外没有不变函数。特别是范畴商简化为一个点。解决这个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“无效下降数据”。堆栈理论和非交换几何都是这种策略的例子。在以前的工作中，PI提出了一种相当不同的方法。这个想法是扩大剧目的职能代数几何毗邻“费马商算子”;新的职能剧目（应被视为算术类似物的非线性微分算子）原来是足够灵活，有时提供有趣的不变量。新的结果几何可以被称为算术微分几何。这是一个交换几何，可以被看作是一个算术模拟的里特-Kolchin微分代数几何。每一个素数都有一个算术微分几何。PI提出的主要猜想是，如果给定一个数域上的代数曲线和一个满足一定自然密度条件的对应关系，则该对应关系对曲线的范畴商在算术微分几何中对几乎所有素数都是非平凡的，当且仅当在复数上，该对应关系允许一个“复解析单值化”。人们可以通过预言在大多数情况下相应的商空间是“有理数”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。目前的研究项目建议继续对这一猜想的工作，将这项工作扩展到高维情况和"偏微分情况"，并利用这一理论与其他理论的相互作用。商空间的构造是几何学中的一个中心问题。在某些自然情况下，在给定的几何形状中，平行物体会表现出基本的病态。这促使人们寻求经典几何的扩展，其中可以避免商病态。该提案提出了一个新的扩展代数几何似乎很适合treatpathologies所产生的算术问题。

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.