0.3. COMPARISON WITH OTHE R THEORIE S xxiii
Here are a few more remarks on differential algebraic jet spaces. As in the case
of smooth manifolds, the fibers of the maps
Jr{X)

Jr~1(X)
are afflne spaces.
Also there is a natural map at the level of K—points V : X(K) *
Jr(X)(K)
naturally induced by the map V : A
1
^ ) = K -+ J
r
(A
1
)(X) =
Kr+1
defined by
V(a) = (a,£a,
...,£ra).
One defines the ring
Or(X)
of 5—polynomial maps on X
to be the ring of all functions / : X{K) » K that can be written as / = / o V
where / G
0(Jr(X))
is a regular function on
Jr(X).
One gets an isomorphism
Or(X)
~
0(Jr(X))
of which the isomorphism in Equation 0.2 is the arithmetic
analogue.
Next we address the following question: are the results of this book arithmetic
analogues of results involving (genuine) differential equations ? In a certain loose
sense this is indeed the case as explained below.
1) The spherical case of our theory here can be loosely viewed as an arith-
metic analogue of some of the classical theory of differential invariants as found, for
instance, in [135], [114].
2) The flat case of our theory here can be loosely viewed as an arithmetic
analogue of the theory of the Manin map [97] and of the differential algebraic
theory we developed in [12], [14], [15].
3) The hyperbolic case of our theory here can be loosely viewed as an arithmetic
analogue of the classical theory of differential relations among modular forms and
of the differential algebraic theory we developed in [16]; cf. also [8].
Let us provide some details.
0.3.2.1. Spherical case. We explain here a simple situation in the classical the-
ory of differential invariants. Start with a group G acting by algebraic auto-
morphisms on a variety X over K (which is usually afflne). Then there is an
induced action of G on the jet spaces Jr(X). The ring of G—invariant func-
tions Or(X)G = 0(Jr(X))G corresponds to a special case of differential invari-
ants in classical theory. A trivial example of this would be the natural action
of G := SL2(C) on the affine plane
A2
= Spec K[x,y] over K; the Wronskian
xy' yx' G K[x, y, x\ y']
0(J1(A1))
is then G—invariant. The spherical case of
our arithmetic theory involves arithmetic analogues of computations of differential
invariants in the above sense.
More generally the classical differential invariant theory deals with groups G
(or simply vector fields V) acting directly on jet spaces Jr(X) (in such a way that
the action does not necessarily come "by functoriality" from an action on X itself).
The situation is described in detail in Olver's book [114] in the setting of jet spaces
in differential geometry.
0.3.2.2. Flat case. As already mentioned our flat theory will involve an arith-
metic analogue of the theory of Manin maps [97], [12], [14], [15]. Let us quickly
review the basics of the latter in the context of differential algebra following [12].
Assume X is an Abelian variety over K of dimension g. Then, by functoriality,
each
Jr(X)
has a natural structure of an algebraic group and is an extension of
A by a vector group (i.e. a sum of copies of the additive group G
a
= Spec K[i\).
By the theory of extensions of Abelian varieties by vector groups there is a surjec-
tive homomorphism
Jr(X)
H onto a vector group H of dimension (r \)g.
Consequently if r 2 we have H ^ 0 hence Hom(Jr(X),Ga) ^ 0. Elements of
the latter composed with V give rise to homomorphisms ip : X(K) » K. These
homomorphisms are called Manin maps. They were discovered by Manin [97] (via
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