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.
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