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