# Birch-Tate conjecture

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

Let $\zeta _ {F}$ denote the 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 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). 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 [a6], the Birch–Tate conjecture was confirmed up to $2$- torsion for Abelian extensions $F$ of $\mathbf Q$.

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

Moreover, [a7] (see the footnote on page 499) together with [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 [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 [a1], [a2].

The Birch–Tate conjecture is related to the Lichtenbaum conjectures [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

 [a1] P.E. Conner, J. Hurrelbrink, "Class number parity" , Pure Math. , 8 , World Sci. (1988) [a2] 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 [a3] M. Kolster, "The structure of the $2$-Sylow subgroup of $K_2(\mathcal{O})$ I" Comment. Math. Helv. , 61 (1986) pp. 376–388 [a4] 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 [a5] 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 [a6] B. Mazur, A. Wiles, "Class fields of abelian extensions of $\QQ$" 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
How to Cite This Entry:
Birch-Tate conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birch-Tate_conjecture&oldid=53302
This article was adapted from an original article by J. Hurrelbrink (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article