DIFFERENTIAL CALCULUS WITH INTEGERS

Abstract. Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is re-placed by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory. The main purpose of these notes is to show how one can develop an arithmetic analogue of differential calculus in which differentiable functions x(t) are replaced by integer numbers n and the derivation operator x 7 → dxdt is replaced by the Fer-mat quotient operator n 7 → n−npp, where p is a prime integer. The Lie-Cartan geometric theory of differential equations (in which solutions are smooth maps) is then replaced by a theory of “arithmetic differential equations ” (in which solutions are integral points of algebraic varieties). In particular the differential invariants of groups in the Lie-Cartan theory are replaced by “arithmetic differential invari-ants ” of correspondences between algebraic varieties. A number of applications to diophantine geometry over number fields and to classical modular forms will be explained. This program was initiated in [11] and pursued, in particular, in [12]-[35]; for an exposition of some of these ideas we refer to the monograph [16]. We shall restrict ourselves here to the ordinary differential case. For the partial differential case we refer to [20, 21, 22, 7]. Throughout these notes we assume familiarity with the basic concepts of algebraic geometry and differential geometry; some of the standard material is being reviewed, however, for the sake of introducing notation, and “setting the stage”. The notes are organized as follows. The first section presents some classical background, the main concepts of the theory, a discussion of the main motivations, and a comparison with other theories. The second section presents a sample of the main results. The third section presents a list of open problems