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