Fermat quotients, correspondences, and uniformization

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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.