PREFACE

IX

the theory of dynamical systems [109], Connes' non-commutative geometry [36],

the theory of Drinfeld modules [52], Dwork's theory [53], Mochizuchi's p—adic

Teichmuller theory [111], Ihara's theory of congruence relations [73] [74], and the

work of Kurokawa, Soule, Deninger, Manin, and others on the "field with one el-

ement" [86], [128], [98], [46]. In Chapter 1 we discuss some algebro-geometric

preliminaries; in particular we discuss analytic uniformization of correspondences

on algebraic curves. In Chapter 2 we discuss our 5—geometric strategy in detail

and we state our main conjectures and a sample of our main results. In Chapters

3, 4, 5 we develop the general theory of arithmetic jet spaces. The correspond-

ing 3 Chapters deal with the global, local, and birational theory respectively. In

Chapters 6, 7, 8 we are concerned with our applications of S—geometry to quotient

spaces: the corresponding 3 Chapters are concerned with correspondences admit-

ting a spherical, flat, or hyperbolic analytic uniformization respectively. Details

as to the contents of the individual Chapters are given at the beginning of each

Chapter. All the definitions of new concepts introduced in the book are numbered

and an index of them is included after the bibliography. A list of references to

the main results is given at the end of the book. Internal references of the form

Theorem x.y, Equation x.y, etc. refer to Theorems, Equations, etc. belonging to

Chapter x (if x ^ 0) or the Introduction (if x — 0). Theorems, Propositions, Lem-

mas, Corollaries, Definitions, and Examples are numbered in the same sequence;

Equations are numbered in a separate sequence. Here are a few words about the

dependence between the various Chapters. The impatient reader can merely skim

through Chapter 1; he/she will need to read at least the (numbered) "Definitions"

(some of which are not standard). Chapters 2-5 should be read in a sequence.

Chapters 6-8 are largely (although not entirely) independent of one another but

they depend upon Chapters 2-5.

Acknowledgments. The author wishes to thank D. Bertrand, P. Deligne,

E. Hrushovski, Y. Ihara, M. Kim, Y. I. Manin, B. Mazur, S. Lang, A. Pillay, B.

Poonen, F. Pop, T. Scanlon, J. Tate, D. Thakur, D. Ulmer, and J. F. Voloch

for encouragement and discussions at various stages of development of these ideas.

While writing this book the author was supported in part by NSF Grant #0096946.

Alexandru Buium

Albuquerque, May 2005