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 .
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].
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 .
|[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://www.encyclopediaofmath.org/index.php?title=Birch%E2%80%93Tate_conjecture&oldid=22123