a construction different from the one above) and were used by him to provide the
first proof of the Mordell conjecture over function fields. Now the strategy pre-
sented above works well in the arithmetic case. The p—jet spaces Jr(Xp) of an
Abelian scheme Xp over Rp continue to be group objects and, although the kernel
of the projection
Xp ceases to be a sum of copies of G
:= Spf Rp[ty,
one can still prove that
Ga) ^ 0 provided r 2. As before elements
of the latter yield 8—functions %jj : X(Rp) Rp which are homomorphisms; these
should be viewed as arithmetic analogues of Manin's maps and play a key role in
our theory in the flat case. Indeed maps of the form ^ r are invariant under the
multiplication by integers N on our Abelian variety, [TV] : Xp » Xp\ this invariance
is the starting point for our use of i\) in our search for invariants in the flat case. Hyperbolic case. We end our discussion of differential algebra by re-
viewing "differential algebraic invariants of isogenics" following [16]; this theory
has an analogue in the arithmetic case that plays a key role in our treatment of
the hyperbolic case. Let us consider the affine line A 1 = Spec K[j] viewed as a
moduli space for elliptic curves over K. For any elliptic curve E over K we let
j(E) G K be its j—invariant. On K A 1 (If) we have then an equivalence relation
called isogeny: two points a±, ci2 G K are called isogenous if there exists an isogeny
of elliptic curves E\ E2 over K such that j(E\) = a\ and ,7(^2) = &2- The
constant field C C K is saturated with respect to isogeny hence so is K\C. Then
one can show [16] that there exists a rational function U(j) G Q(j) such that if
(0.24) xUJ'J"J'") •=
e K(j)[f, ur\j"J'"}
then the function
K\C - K, a ^
, 6a,
52a, 53a)
is constant on isogeny classes. (The fraction in Equation 0.24 is, of course, the
classical Schwarzian operator.) We refer to [16] for more on this and for a higher
dimensional generalization of this. It turns out that the theory behind the function
X above has an arithmetic analogue which leads to what we call 5—modular forms;
cf. [21], [23], [24]. These will be used heavily in this book.
0.3.3. Difference algebra. In the Ritt-Cohn difference algebra [35] one starts
with a field K equipped with a ring endomorphism (j) : K K. For our discussion
here we will allow K to have arbitrary characteristic. For any variety X over K
one can then define a projective system of non-singular varieties
... -+ J;(X) - J;~\X) -+... -+ j°{x) = x
which can be called the difference jet spaces of X. If X is affine given by
X = SpecK[T]/{F)
where T is a tuple of indeterminates and F is a tuple of elements in K[T] then
J1{X) is, by definition, the spectrum
(0.25) j;(X) := Spec K[T, T*, T**,..., T^]/{F, /F,
02F,..., /rF)
where T^,T^ ,... are new tuples of variables and
0 : K[T, T*,...,
- K[T, T*,..., T^}
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