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
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
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
vii) For R a ring with unit,
denotes the multiplicative group of invertible ele-
ments in R.
viii) All homomorphisms between rings with unit are supposed to respect the
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
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
xii) If σ : R → R is an automorphism of R, then
denotes the R-algebra
−→ A. If M is an R-module, then we put M
:= M ⊗R
an Rσ-module as well as an R-module.
xiii) All central simple algebras are assumed to be finite dimensional over a
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.