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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
.
References
[a1] | P.E. Conner, J. Hurrelbrink, "Class number parity" , Pure Math. , 8 , World Sci. (1988) |
[a2] | J. Hurrelbrink, "Class numbers, units, and ![]() ![]() |
[a3] | M. Kolster, "The structure of the ![]() ![]() |
[a4] | M. Kolster, "A relation between the ![]() |
[a5] | S. Lichtenbaum, "Values of zeta functions, étale cohomology, and algebraic ![]() ![]() |
[a6] | B. Mazur, A. Wiles, "Class fields of abelian extensions of ![]() |
[a7] | A. Wiles, "The Iwasawa conjecture for totally real fields" Ann. of Math. , 131 (1990) pp. 493–540 |
Birch-Tate conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birch-Tate_conjecture&oldid=11217