Birch-Tate conjecture

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