0.3. COMPARISON WIT H OTHE R THEORIE S xix

for this purpose. There is, for instance, a Galois theoretic method that is sufficient

to handle, in particular, the spherical case. Another method, especially useful in

the flat and hyperbolic case, is close in spirit to that used by Hilbert in his original

approach to invariant theory and is based on the existence of a certain system of

(usual) partial differential operators acting on appropriate rings of invariants. In

our case the partial differential operators in question will live on arithmetic jet

spaces and will be derived by analogy with classical mechanics.

0.3. Compariso n wit h other theories

0.3.1. Differential equations o n s m o o t h manifolds. Many of the 5— geo-

metric concepts to be introduced and studied in this book are arithmetic analogues

of concepts related to differential equations on smooth manifolds derived from clas-

sical mechanics. Let us quickly review here some of these smooth manifold concepts.

Let M be a smooth manifold (i.e. a C°° real manifold) equipped with a sub-

mersion 7r : M —•» R. For each t G R we think of

7r-1(£)

as the "configuration

space" at time t. (The map M —• R should be viewed as the object whose arith-

metic analogues are the morphisms of schemes, X — Spec F , defining varieties X

over number fields F.) For any smooth map / : R — » M we denote by J Q ( / )

t n e

r—jet of / at 0. We define the jet space of M (relative to n) by

(0.6)

Jr(M)

:= { JQ (/) | / : R - • M smooth, TT O / : R -» R a translation}.

Then

Jr(M)

has a natural structure of smooth manifold. Any smooth section

s : R — M of TT : M — R lifts to a smooth map pr s : R —

Jr{M)

defined by

(pr s){t) := J Q ( s o r

t

) ,

where rt : R — R is the translation rt{u) := u + t. The smooth functions L :

Jr(M)

— • R can be thought of as (time dependent) Lagrangians. Alternatively

we may think of such smooth functions L as differential equations on M. The

natural projections

Jr(M)

—

Jr~1(M)

are fiber bundles with fiber R

n

, where

dim M = n + 1.

Next, in order to simplify our discussion, we will work in coordinates. So we

let M :=H x R

n

with coordinates (£, x) = (t, xi , ...,xn) and we let n : M — R be

the first projection. Then we have a natural identification

Jr(M)

= R x R

n

(

r + 1

)

sending the r— jet JQ (/) of a map /

:

R —* M, f(t) = (t + to, x(t)), into the tuple

H°.!«» £•»)•

If 5 : R - R x R

n

, s(£) = (£, x(t)), is a section of n then p r s : R - R x R"(r+i)

is given by

(0.7) ( p r , ) ( t ) = ^ , x ( t ) , ^ ( t ) , . . . , ^ ( t ) ) .

Cf. [114], p. 96. We denote by (t,x,x',...,x^) the coordinates on R x R

n

(

r + 1

) ,

where

x1', ...,x^r^

are n—tuples of variables; for each index i the variables x'^x",...

are the variables conjugate toxi. A (time dependent) Lagrangian is simply a smooth

function

(0.8)

L(t,x,x',...,x(r))

on R x R

n

(

r + 1

) . Below is a list of key concepts related to the above context.