View source for Birch-Tate conjecture

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Let  $  {\mathcal O} _ {F} $
be the ring of integers of an algebraic number [[Field|field]]  $  F $(
cf. also [[Algebraic number|Algebraic number]]). The Milnor  $  K $-
group  $  K _ {2} ( {\mathcal O} _ {F} ) $, 
which is also called the tame kernel of  $  F $, 
is an [[Abelian group|Abelian group]] of finite order.

Let  $  \zeta _ {F} $
denote the [[Dedekind zeta-function|Dedekind zeta-function]] of  $  F $. 
If  $  F $
is totally real, then  $  \zeta _ {F} ( - 1 ) $
is a non-zero rational number, and the Birch–Tate conjecture is about a relationship between  $  \zeta _ {F} ( - 1 ) $
and the order of  $  K _ {2} ( {\mathcal O} _ {F} ) $.

Specifically, let  $  w _ {2} ( F ) $
be the largest natural number  $  N $
such that the [[Galois group|Galois group]] of the cyclotomic extension over  $  F $
obtained by adjoining the  $  N $
th roots of unity to  $  F $, 
is an elementary Abelian  $  2 $-
group (cf. [[P-group| $  p $-
group]]). Then  $  w _ {2} ( F ) \cdot \zeta _ {F} ( - 1 ) $
is a rational integer, and the Birch–Tate conjecture states that if  $  F $
is a totally real number field, then

$$ 
\# K _ {2} ( {\mathcal O} _ {F} ) = \left | {w _ {2} ( F ) \cdot \zeta _ {F} ( - 1 ) } \right | .
$$

A numerical example is as follows. For  $  F = \mathbf Q $
one has  $  w _ {2} ( \mathbf Q ) = 24 $, 
$  \zeta _ {\mathbf Q} ( - 1 ) = - {1 / {12 } } $; 
so it is predicted by the conjecture that the order of  $  K _ {2} ( \mathbf Z ) $
is  $  2 $, 
which is correct.

What is known for totally real number fields  $  F $?

By work on the main conjecture of Iwasawa theory [[#References|[a6]]], the Birch–Tate conjecture was confirmed up to  $  2 $-
torsion for Abelian extensions  $  F $
of  $  \mathbf Q $.

Subsequently, [[#References|[a7]]], the Birch–Tate conjecture was confirmed up to  $  2 $-
torsion for arbitrary totally real number fields  $  F $.

Moreover, [[#References|[a7]]] (see the footnote on page 499) together with [[#References|[a4]]], also the  $  2 $-
part of the Birch–Tate conjecture is confirmed for Abelian extensions  $  F $
of  $  \mathbf Q $.

By the above, all that is left to be considered is the  $  2 $-
part of the Birch–Tate conjecture for non-Abelian extensions  $  F $
of  $  \mathbf Q $. 
In this regard, for extensions  $  F $
of  $  \mathbf Q $
for which the  $  2 $-
primary subgroup of  $  K _ {2} ( {\mathcal O} _ {F} ) $
is elementary Abelian, the  $  2 $-
part of the Birch–Tate conjecture has been confirmed [[#References|[a3]]].

In addition, explicit examples of families of non-Abelian extensions  $  F $
of  $  \mathbf Q $
for which the  $  2 $-
part of the Birch–Tate conjecture holds, have been given in [[#References|[a1]]], [[#References|[a2]]].

The Birch–Tate conjecture is related to the Lichtenbaum conjectures [[#References|[a5]]] for totally real number fields  $  F $. 
For every odd natural number  $  m $, 
the Lichtenbaum conjectures express, up to  $  2 $-
torsion, the ratio of the orders of  $  K _ {2m }  ( {\mathcal O} _ {F} ) $
and  $  K _ {2m+1 }  ( {\mathcal O} _ {F} ) $
in terms of the value of the zeta-function  $  \zeta _ {F} $
at  $  - m $.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.E. Conner,   J. Hurrelbrink,   "Class number parity" , ''Pure Math.'' , '''8''' , World Sci.  (1988)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Hurrelbrink,   "Class numbers, units, and $K_2$"  J.F. Jardine (ed.)  V. Snaith (ed.) , ''Algebraic K-theory: Connection with Geometry and Topology'' , ''NATO ASI Ser. C'' , '''279''' , Kluwer Acad. Publ.  (1989)  pp. 87–102</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Kolster,   "The structure of the $2$-Sylow subgroup of $K_2(\mathcal{O})$ I"  ''Comment. Math. Helv.'' , '''61'''  (1986)  pp. 376–388</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Kolster,   "A relation between the $2$-primary parts of the main conjecture and the Birch–Tate conjecture"  ''Canad. Math. Bull.'' , '''32''' :  2  (1989)  pp. 248–251</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Lichtenbaum,   "Values of zeta functions, étale cohomology, and algebraic K-theory"  H. Bass (ed.) , ''Algebraic K-theory II'' , ''Lecture Notes in Mathematics'' , '''342''' , Springer  (1973)  pp. 489–501</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Mazur,   A. Wiles,   "Class fields of abelian extensions of $\QQ$"  ''Invent. Math.'' , '''76'''  (1984)  pp. 179–330</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Wiles,   "The Iwasawa conjecture for totally real fields"  ''Ann. of Math.'' , '''131'''  (1990)  pp. 493–540</TD></TR></table>