xviii INTRODUCTION

projective system of formal schemes

(o.i) ... -+ r{xp) - r~\xp) -+... -+

J°(XP)

= xp

such that for each r the ring Or{Xp) of £—functions of order r on Xp(Rp)

identifies with the ring

0(Jr(Xp))

of global functions on

Jr(Xp):

(0.2) Or(Xp) ~ 0(Jr(Xp)).

The formal scheme

Jr(Xp)

should be viewed intuitively as an "arithmetic jet space"

and will be referred to as the p—jet space of order r of Xp. It is an arithmetic ana-

logue of the jet space of a manifold (relative to a submersion) in differential geom-

etry (cf. Equation 0.6) and, in particular, it carries certain structures reminiscent

of structures in classical mechanics. The elements of the rings in Equation 0.2 can

then be viewed intuitively as "arithmetic differential equations" in the same way in

which functions on jet spaces in differential geometry are interpreted as "differential

equations". The above construction reduces the study of

u5—geometry"

of Xp to

the study of (usual) algebraic geometry of

Jr(Xp).

Actually the construction of

p—jet spaces is quite easy to explain. Assume, for simplicity, that Xp is affine given

by

(0.3) Xp = SpecRp[T}/(F),

where T is a tuple of indeterminates and F is a tuple of elements in Rp [T]. Assume

furthermore, for simplicity, that the components of F have coefficients in Zp. Then

Jr(Xp) is, by definition, the formal spectrum

(0.4)

Jr(Xp)

:= Spf

Rp[T,T',T",-,T^Y/(F,SF,S2F,...,6rF)

where " means

up—adic

completion",

T',71",...

are new tuples of variables and

5F,

52F,...

are defined by the formulae

(0.5)

{SFKT,r):=F{TP+Pr)-F{T)P,

P

(52F)(T,T',T") := (SF)(TP+Pr^r)p+PT") - ((SF)(TX))P^

p

etc. The polynomials 5F,

52F,...

should be viewed as arithmetic analogues of "iter-

ated total derivatives" of F and the construction of our "arithmetic jet spaces"

is then analogous to that of "differential algebraic jet spaces" to be discussed

presently; cf. Equation 0.20 below. Also, like in the case of the latter, and as

in the case of differential geometry, the fibers of the maps

Jr(Xp)

—

Jr~1(Xp)

are

(p—adic completions of) affine spaces.

Having reduced our problems to problems about p—jet spaces the next step is

to construct, in each of the spherical, flat, and hyperbolic cases, some remarkable

5—invariants. In the spherical case this will be elementary. In the flat case the

construction will be essentially based on our theory of 5—characters [17]; the latter

are arithmetic analogues of the Manin maps in [97]. In the hyperbolic case the

construction will be based on our theory of isogeny covariant S—modular forms

[21], [23], [24]. We will develop the theories of S—characters and S—modular forms

ab initio in this book; we will partly follow the above cited papers and then we

will further develop these theories up to a point where we can use them for our

applications.

Finally we will need methods to prove that all S—invariants essentially occur

via the constructions mentioned above. There will be a number of methods used