Differential calculus with integers

Abstract

Abstract. Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced 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 → dx dt is replaced by the Fermat quotient operator n → n−n p p , 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 invariants" 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.Acknowledgement. The author is indebted to HIM for support during part of the semester on Algebra and Geometry in Spring 2013. These notes are partially based on lectures given at the IHES in Fall

Rigid differentially closed fields

“…But if x ∈ X a \ k alg , then k⟨x⟩ = k(x) is a transcendence degree 1 extension. But by results of Buium [2], Manin kernels, or anything nonorthogonal to one, give rise to extensions of transcendence degree at least 2. Thus X a is trivial.…”

“…Our main tool will be the strongly minimal sets known as Manin kernels of elliptic curves. Manin kernels arose in Manin's proof [10] of the Mordell conjecture for function fields in characteristic zero and were central to both Buium's [2] and Hrushovski's [8] proofs of the Mordell-Lang conjecture for function fields in characteristic zero. The model theoretic importance of Manin kernels was developed in the beautiful unpublished preprint of Hrushovski and Sokolović [9].…”

“…We build a chain 2 If we can do that we will have guaranteed that there are no automorphisms of K . Let K 0 = ‫ޑ‬ diff .…”

“…The next lemma shows that we have the necessary bounds. 2 Building the enumeration takes a bit more bookkeeping in this case. Let d 0,0 , d 0,1 , .…”

A Zariski--Nagata theorem for smooth ℤ-algebras

“…Rather than using perfectoid techniques, our generalization of Zariski-Nagata makes use of a different arithmetic notion of derivative, the notion of a p-derivation, defined by Joyal [Joy85] and Buium [Bui95] independently. From a commutative algebra point of view, p-derivations are rather exotic maps from a ring to itself -in particular, they are not even additive -but they do have many applications to arithmetic geometry, such as in [Bui05,Bui15,Bor09]. To the best of our knowledge, this is the first application of p-derivations to commutative algebra.…”