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
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