XIV

INTRODUCTION

operators attached to various prime numbers p one is led to what can be called

arithmetic differential algebra (and its corresponding geometry). The case when 5

is a p—difference operator seems to lead to a less interesting theory. Of the above

4 types of 5, p—derivations are the only ones that do not vanish identically on the

integers. This makes p—derivations especially suited for arithmetic applications; so

we shall exclusively be interested, in this book, in arithmetic differential algebra.

The study of arithmetic differential algebra and its associated geometry was begun

in our paper [17] and developed in a series of papers [18]-[28], [2], [3], [71]. Earlier

papers (cf. especially papers by Joyal [79] and Ihara [75]) contain indications that

Fermat quotients should be viewed as analogues of derivations. However note that

there is an important difference between our approach and the one proposed by

Ihara; cf. the remarks surrounding Equation 0.31 below. Arithmetic differential

algebra can be viewed as an "arithmetic" analogue of the Ritt-Kolchin differential

algebra and also as a "regularized" analogue of the Cohn difference algebra; indeed

arithmetic differential algebra can be viewed as obtained from difference algebra by

adjoining to it certain divergent series (cf. Remark 2.6 below).

As a matter of terminology note that, in the Ritt-Kolchin differential algebra

one uses the symbol 5 as an abbreviation for "differential". By analogy with the

tradition in the Ritt-Kolchin theory we will use the symbol S as an abbreviation for

"arithmetic differential"; in particular arithmetic differential algebra will be referred

to as S—algebra while its "associated" geometry, which can be called arithmetic

differential geometry, will be referred to as 5—geometry. There will be no danger of

confusion with differential algebra and difference algebra terminology because no

use of the latter two types of algebra will be made in this book.

The main purpose of this book is to first develop some of the basic elements of

£—geometry and then to construct (and study) interesting categorical quotients, in

S—geometry, for correspondences whose categorical quotient in algebraic geometry

is trivial. A conjecture will emerge to the effect that the £—geometric picture is

interesting if and (essentially) only if our correspondences admit a complex analytic

uniformization. Below we give a rough outline of our theory; the details of the

theory will be explained in Chapter 2.

0.2. Rough outline of the theory

0.2.1. Background and notation. We start with a number field F (always

assumed of finite degree over Q). Fix a finite place p of F which is unramified

over Q; later we will vary p. We identify p with a maximal ideal in the ring of

integers Op of F. Let p = char O/p. Let Op be the localization of Op at p,

consider its completion Op, and let

Opr

be the maximum unramified extension of

Op (obtained by adjoining to Op all the roots of unity of order prime to p in an

algebraic closure of its fraction field). Then

OpT

is a discrete valuation ring with

maximal ideal generated by p. Consider the completion of Op^ which we denote by

RP ••=

(ouPrr-

The ring Rp has a simple well known structure: any element of it can be represented

uniquely as a series Y^LQ

CiP%

where Q are roots of unity of order prime to p or 0.

The ring Rp has a unique automorphism / inducing the p—power Frobenius map

on the residue field kp := Rp/pRp; it is given by (pQZ^0CiPl) — SSoClV - For