10 INTRODUCTION

• Abelian varieties and homomorphisms between them.

– The dual of an abelian variety A is denoted

At.

– For an abelian variety A over a field K and a prime not divisible by

char(K), upon choosing a separable closure Ks of K (often understood

from context) the -adic Tate module T (A) denotes lim

← −

A[

n](Ks)

and

V (A) denotes Q ⊗Z T (A).

– For any abelian varieties A and B over a field K, Hom(A, B) denotes

the group of homomorphisms A → B over K, and

Hom0(A,

B) denotes

Q ⊗Z Hom(A, B).

(Since Hom(A, B) → Hom(AK , BK) is injective, Hom(A, B) is a finite

free Z-module since the same holds over K by [82, §19, Thm. 3].)

– When B = A we write End(A) and

End0(A)

respectively, and call

End0(A)

the endomorphism algebra of A (over K). The endomorphism algebra

End0(A)

is an invariant which only depends on A up to isogeny over K,

in contrast with the endomorphism ring End(A).

– We write A ∼ B to denote that abelian varieties A and B over K are

K-isogenous.

– To avoid any possible confusion with notation found in the literature, we

emphasize that what we call Hom(A, B) and

Hom0(A,

B) are sometimes

denoted by others as HomK (A, B) and HomK

0

(A, B).11

• Adeles and local fields.

– We write AL to denote the adele ring of a number field L, AL,f to denote

the factor ring of finite adeles, and A and Af in the case L = Q.

– If v is a place of a number field L then Lv denotes the completion of L

with respect to v; OL,v denotes the valuation ring OLv of Lv in case v is

non-archimedean, with residue field κv whose size is denoted qv.

– For a place w of Q we define Lw := Qw ⊗Q L =

v|w

Lv, and in case w

is the -adic place for a prime we define OL, := Z ⊗Z OL =

v|

OL,v.

• Class field theory and reciprocity laws.

– The Artin maps of local and global class field theory are taken with

the arithmetic normalization, which is to say that local uniformizers are

carried to arithmetic Frobenius

elements.12

– recL : AL

×/L×

→

Gal(Lab/L)

denotes the arithmetically normalized glob-

al reciprocity map for a number field L.

– The composition of AL

×

AL

×/L×

with recL is denoted rL.

– For a non-archimedean local field F we write rF : F

×

→ Gal(F

ab/F

) to

denote the arithmetically normalized local reciprocity map.

• Frobenius and Verschiebung.

– For a commutative group scheme N over an Fp-scheme S, N

(p)

denotes

the base change of N by the absolute Frobenius endomorphism of S. The

relative Frobenius homomorphism is denoted FrN/S : N → N

(p),

and the

11with

the notation Hom(A, B) and

Hom0(A,

B) then reserved to mean the analogues for

A

K

and B

K

over K, or equivalently for AKs and BKs over Ks (see Lemma 1.2.1.2).

12Recall

that for a non-archimedean local field F with residue field of size q, an element of

Gal(Fs/F ) is called an arithmetic (resp. geometric) Frobenius element if its effect on the residue

field of Fs is the automorphism x →

xq

(resp. x →

x1/q);

this automorphism of the residue

field is likewise called the arithmetic (resp. geometric) Frobenius automorphism. We choose the

arithmetic normalization of class field theory so that uniformizers correspond to Frobenius endo-

morphisms of abelian varieties in the Main Theorem of Complex Multiplication.