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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  " J.F. Jardine (ed.) V. Snaith (ed.) , Algebraic  -theory: Connection with Geometry and Topology , NATO ASI Ser. C , 279 , Kluwer Acad. Publ. (1989) pp. 87–102 |
| [a3] | M. Kolster, "The structure of the  -Sylow subgroup of  I" Comment. Math. Helv. , 61 (1986) pp. 376–388 |
| [a4] | M. Kolster, "A relation between the  -primary parts of the main conjecture and the Birch–Tate conjecture" Canad. Math. Bull. , 32 : 2 (1989) pp. 248–251 |
| [a5] | S. Lichtenbaum, "Values of zeta functions, étale cohomology, and algebraic  -theory" H. Bass (ed.) , Algebraic  -theory II , Lecture Notes in Mathematics , 342 , Springer (1973) pp. 489–501 |
| [a6] | B. Mazur, A. Wiles, "Class fields of abelian extensions of  " Invent. Math. , 76 (1984) pp. 179–330 |
| [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=22123