6 1. PRELIMINARIES FROM ALGEBRAIC GEOMETR Y

the affine space over R and by

(1.9) G

0

: = ( A \ + ) , G

m

:=(5peci?[T,T-

1

],-)

the additive and multiplicative group schemes over R respectively. An elliptic curve

over R is a smooth proper group scheme over R of relative dimension one with

irreducible geometric fibers. (See the discussion below for a review of smoothness.)

If X is a Noetherian scheme and P G X is a closed point we denote by

(1.10) X, Xfor

the p—adic completion of X and the completion of X at P respectively.

Recall Grothendieck's existence theorem saying in particular that if X and Y

are projective schemes over a p—adically complete ring S then the map

Homs(X,Y) - Homs{X,Y)

is a bijection.

Let X be a scheme (resp. a formal scheme) and let G — X be a group scheme

over X (resp. a group formal scheme over X). A principal homogeneous space

under G is a morphism Y — X of schemes (resp. a formal schemes) together with

an action m : G xx Y — Y such that the induced morphism

m x pr2 : G xxY -+Y xxY

is an isomorphism. A principal homogeneous space is said to be trivial if it has a

section; equivalently if it is isomorphic to the principal homogeneous space G — X.

By a Galois cover of schemes we understand a finite morphism of schemes

n : Y - X such that

{ir*Oy)T

= Ox, where V := Aut(Y/X) is the group of

X—automorphisms of Y; T is referred to as the Galois group of the cover.

1.1.4. Derivations and smoothness. If B is an A—algebra and C is a

B—algebra with structure homomorphism / : B — C we denote by DerA(B,C)

the B—module of A—derivations D : B — C relative to / , by which we mean

A—module homomorphisms satisfying

D(xy) = f{x) • D(y) + f(y) • D(x),

for x,y G B. If A is a sheaf of rings on a topological space, B is a sheaf of

A—algebras, and C is a sheaf of B—algebras we denote by Der^(B,C) the group of

.4—derivations B — C and by Ver^(B, C) the sheaf of C—modules of A—derivations

B^C.

We recall here the basic definitions and properties of smoothness we are using

in this book; cf. [10], Chapter 2. Let A be a ring and B a finitely generated

A—algebra. One says B is smooth (resp. etale) over A if for any A—algebra C and

any ideal I C C with

I2

= 0 the canonical map

HomA(B,C) - HomA(B,C/I)

is surjective (resp. bijective). A scheme X over A is called smooth (resp. etale) if

it can be covered with open affine schemes Spec B with B finitely generated and

smooth (resp. etale) over A. A morphism / : X — • Y of schemes is called smooth

(resp. etale) if Y can be covered by affine open schemes U — Spec A such that

f~x(U)

are smooth (resp. etale) over A. A morphism X — Y of finite type of

Noetherian schemes is smooth if and only if it is flat and all geometric fibers are

regular. If Spec B — Spec A is of finite type and smooth (resp. etale) then B is

smooth (resp. etale) over A. If a scheme X over A is smooth then each point in X