introduction 11

Notation and conventions

We follow standard notation and conventions from algebra, algebraic number the-

ory, and algebraic geometry. More precisely:

i) We write , , , , and for the sets of natural numbers, integers, rational

numbers, real numbers, and complex numbers, respectively.

ii) We say that a function f is O(g), for B → ∞, if there exists a constant C ∈

such that f(B) ≤ Cg(B), for B suﬃciently large. Instead of f = O(g), we also

write f g or g f .

iii) For a group G and elements σ1, . . . , σn ∈ G, we denote the subgroup generated

by σ1, . . . , σn by σ1, . . . , σn ⊆ G.

If G is abelian, then Gn ⊆ G is the subgroup consisting of all elements of torsion

dividing n.

iv) If a group G operates on a set M , then M G denotes the invariants. We write

M σ instead of M σ .

v) All rings are assumed to be associative.

vi) If R is a ring, then

Rop

denotes the opposite ring. I.e., the ring that coincides

with R as an abelian group, but in which one has xy = z when one had yx = z

in R.

vii) For R a ring with unit,

R∗

denotes the multiplicative group of invertible ele-

ments in R.

viii) All homomorphisms between rings with unit are supposed to respect the

unit elements.

ix) By a field, we always mean a commutative field. I.e., a commutative ring with

unit, every non-zero element of which is invertible. If K is a field, then Ksep and K

denote a fixed separable closure and a fixed algebraic closure, respectively.

x) A ring with unit, every non-zero element of which is invertible, is called a

skew field.

xi) If R is a commutative ring with unit, then an R-algebra is always understood

to be a ring homomorphism j : R → A, the image of which is contained in the

center of A. An R-algebra j : R → A is denoted simply by A when there seems

to be no danger of confusion. An R-algebra being a skew field is also called a

division algebra.

xii) If σ : R → R is an automorphism of R, then

Aσ

denotes the R-algebra

R

σ

−→ R

j

−→ A. If M is an R-module, then we put M

σ

:= M ⊗R

Rσ.

M

σ

is

an Rσ-module as well as an R-module.

xiii) All central simple algebras are assumed to be finite dimensional over a

base field.

xiv) For K a number field, we write OK to denote the ring of integers in K.

If ν ∈ Val(K) is a non-Archimedean valuation, then the ν-adic completion of K is

denoted by Kν and its ring of integers by OKν .

In the particular case that K = , we denote by νp the normalized p-adic valuation

corresponding to a prime number p.

xv) For R a commutative ring, we denote by Spec R the aﬃne scheme constituted

by its spectrum.