Let
be the ring of integers of an
algebraic number
field
(cf. also
Algebraic number).
The
Milnor
-group
,
which is also called the
tame kernel
of
,
is an
Abelian group
of finite order.
Let
denote the
Dedekind zeta-function
of
.
If
is totally real, then
is a non-zero rational number, and the Birch–Tate conjecture is about
a relationship between
and the order of
.
Specifically, let
be the largest natural number
such that the
Galois group
of the cyclotomic extension over
obtained by adjoining the
th
roots of unity to
,
is an elementary Abelian
-group
(cf.
-group).
Then
is a rational integer, and the
Birch–Tate conjecture
states that if
is a totally real number field, then
A numerical example is as follows. For
one has
,
;
so it is predicted by the conjecture that the order of
is
,
which is correct.
What is known for totally real number fields
?
By work on the main conjecture of
Iwasawa theory
[a6],
the
Birch–Tate conjecture was confirmed up to
-torsion
for Abelian extensions
of
.
Subsequently,
[a7],
the Birch–Tate conjecture was confirmed
up to
-torsion
for arbitrary totally real number fields
.
Moreover,
[a7]
(see the footnote on page 499) together with
[a4],
also the
-part
of the Birch–Tate conjecture is confirmed for Abelian extensions
of
.
By the above, all that is left to be considered is the
-part
of the Birch–Tate conjecture for non-Abelian extensions
of
.
In this regard, for extensions
of
for which the
-primary
subgroup of
is elementary Abelian, the
-part
of the Birch–Tate conjecture has been confirmed
[a3].
In addition, explicit examples of families of non-Abelian extensions
of
for which the
-part
of the Birch–Tate conjecture holds, have been given in
[a1],
[a2].
The Birch–Tate conjecture is related to the
Lichtenbaum conjectures
[a5]
for totally real number fields
.
For every odd natural number
,
the Lichtenbaum conjectures express, up to
-torsion,
the ratio of the orders of
and
in terms of the value of the zeta-function
at
.