Vll l

PREFACE

Our results will suggest a general conjecture according to which the quotient of a

curve (defined over a number field) by a correspondence is non-trivial in £—geometry

for almost all primes p if and only if the correspondence has an "analytic uniformiza-

tion" over the complex numbers. Then the 3 classes of examples above correspond

to spherical, flat, and hyperbolic uniformization respectively.

Material included. The present book follows, in the initial stages of its

analysis, a series of papers written by the author [17]-[28]. A substantial part of

this book consists, however, of material that has never been published before; this

includes our Main Theorems stated at the end of Chapter 2 and proved in the

remaining Chapters of the book. The realization that the series of papers [17]-[28]

consists of pieces of one and the same puzzle came relatively late in the story and

the unity of the various parts of the theory is not easily grasped from reading the

papers themselves; this book is an attempt at providing, among other things, a

linear, unitary account of this work. Discussed are also some of the contributions

to the theory due to C. Hurlburt [71], M. Barcau [2], and K. Zimmerman [29].

Material omitted. A problem that was left untouched in this book is that of

putting together, in an adelic picture, the various 5—geometric pictures, as p varies.

This was addressed in our paper [26] where such an adelic theory was developed

and then applied to providing an arithmetic differential framework for functions of

the form (p, a) i— c(p, a), p prime, a G Z, where L(a, s) = J2n

c(n a

)

n _ s a r e

various

families of L—functions parameterized by a. Another problem not discussed in this

book is that of generalizing the theory to higher dimensions. A glimpse into what

the theory might look like for higher dimensional varieties can be found in [17] and

[3]. Finally, we have left aside, in this book, some of the Diophantine applications

of our theory such as the new proof in [18] of the Manin-Mumford conjecture about

torsion points on curves and the results in [21], [2] on congruences between classical

modular forms.

Prerequisites. For most of the book, the only prerequisites are the basic

facts of algebraic geometry (as found, for instance, in R. Hartshorne's textbook

[66]) and algebraic number theory (as found, for instance, in Part I of S. Lang's

textbook [87]). In later Chapters more background will be assumed and appropriate

references will be given. In particular the last Chapter will assume some familiarity

with the p—adic theory of modular and Shimura curves. From a technical point of

view the book mainly addresses graduate students and researchers with an interest

in algebraic geometry and / or number theory. However, the general theme of the

book, its strategy, and its conclusions should appeal to a general mathematical

audience.

Plan of the book. We will organize our presentation around the motivating

"quotient space" theme. So quotient spaces will take center stage while "arithmetic

jet spaces" and the corresponding analogies with the theory of ordinary differential

equations will appear as mere tools in our proofs of S—geometric theorems. Accord-

ingly, the Introduction starts with a general discussion of strategies to construct

quotient spaces and continues with a brief outline of our S—geometric theory. We

also include, in our Introduction, a discussion of links, analogies, and / or dis-

crepancies between our theory and a number of other theories such as: differential

equations on smooth manifolds [114], the Ritt-Kolchin differential algebra [117],

[84], [32], [13], the difference algebraic work of Hrushovski and Chatzidakis [69],