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
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