XXIV

INTRODUCTION

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

Jr(Xp)

— • Xp ceases to be a sum of copies of G

a

:= Spf Rp[ty,

one can still prove that

Hom(Jr(Xp),

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.

0.3.2.3. 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'") •=

2J'J'"4[/)2{f)2

+

(f)2U(j)

e K(j)[f, ur\j"J'"}

then the function

K\C - K, a ^

x

(

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*,...,

T^"1)

- K[T, T*,..., T^}