0.2. ROUGH OUTLINE O F TH E THEORY

xvn

Here, by X c admitting an analytic uniformization, we understand that there

exist morphisms of correspondences in the category of (open) Riemann surfaces (i.e.

in the category of complex analytic manifolds of dimension one), (*,, V) : X c —» Y c ,

(TT,7r) : Yfc —» Y c , where L, I are open immersions, TT, TT are (possibly infinite)

Galois covers unramified above X , X , and Y'c — ( S , S , T I , T 2 ) , with S a simply

connected Riemann surface and TI, T2 automorphisms of S. We always assume here

that the Galois groups of TT and TT are admissible in the sense that they act prop-

erly discontinuously on S and are either finite (in case S = P 1 ) or infinite (in case

S = C) or of finite covolume (if S = H , the upper half plane) respectively. Accord-

ing; to whether S i s P 1 , C,o r H there are three classes of correspondences admitting

analytic uniformizations which will be called spherical, flat, and hyperbolic corre-

spondences. Analytically uniformizable correspondences can be completely classi-

fied in algebraic terms; cf. Chapter 1. Up to an appropriate equivalence relation

we have, very roughly speaking, the following description. (N.B. The actual situ-

ation is slightly more complicated.) The spherical uniformizable correspondences

are, essentially, of the form ( P ^ P 1 , ^ , ^ o r) where TT : P 1 — » P 1 is a finite Galois

cover and r : P 1 —» P 1 is an isomorphism; the categorical quotient in the category

of sets of such a correspondence is the same as the quotient of P

1

by the action of

the group (T, r) generated by r and the Galois group, T, of TT. Of course the group

( r , r ) is generally infinite. Flat uniformizable correspondences are, essentially, of

the form (P 1 , P 1 , ai , cr2) where r^ are either multiplicative functions t \-^ tdl or

Chebyshev polynomials, or Lattes functions [109] (the latter being induced by en-

domorphisms of elliptic curves E via isomorphisms E/(j) ~ P 1 where 7 : E — E

is an automorphism). The hyperbolic uniformizable correspondences (with infinite

orbits) are, essentially, Hecke correspondences on modular or Shimura curves (that

are classically described in terms of quaternion algebras over totally real fields). So

the "if" part of our conjecture essentially says that our 5—geometric theory gives

a rich picture in all the above examples; the "only if" part of the conjecture says

that the above examples are, essentially, the only ones for which our S—geometric

theory gives a rich picture.

0.2.5. Results. Our main results show that the "if" part of the conjecture

above holds (under some mild assumptions) in the spherical case, in the flat case,

and in the "rational hyperbolic" case. (Here the "rational hyperbolic" case refers

to the case when S = H and the quaternion algebra describing the situation has

center Q.) For correspondences X admitting such a uniformization we will study the

"geometry" and "cohomology" of the quotients Xs/cr§. In particular we will show

that the quotients Xs/0-5 tend to behave like "rational varieties" and the quotient

maps X§ — Xs/as tend to look like "pro-finite covers" whose Galois properties we

shall study. The "only if" part of the conjecture is much more mysterious. We will

be able to prove a local analogue of the "only if" part of our conjecture. Also we will

prove a global result along the "only if" direction saying (roughly speaking) that

if X5/0-5 is "sufficiently non-trivial" for almost all places p then X c is critically

finite (in the sense of complex dynamics) and X

p

0 kp has a generically trivial

pluricanonical bundle. The latter property will allow us to prove a version of the

"only if" part of the conjecture in the "dynamical system case".

0.2.6. Proofs. Here are a few words about the strategy of our proofs. The

first move is to attach to any smooth scheme of finite type Xp over Rp = Z!^r a