Vll l
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
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
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],
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