Birch–Tate conjecture

The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.

Statement

In algebraic K-theory, the group K₂ is defined as the center of the Steinberg group of the ring of integers of a number field F. K₂ is also known as the tame kernel of F. The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function $\zeta _{F}$ . More specifically, let F be a totally real number field and let N be the largest natural number such that the extension of F by the Nth root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that

\#K_{2}=|N\zeta _{F}(-1)|.

Progress on this conjecture has been made as a consequence of work on Iwasawa theory, and in particular of the proofs given for the so-called "main conjecture of Iwasawa theory."

The problem remains unsolved as of 2011.

J. T. Tate, Symbols in Arithmetic, Actes, Congrès Intern. Math., Nice, 1970, Tome 1, Gauthier–Villars(1971), 201–211

Hurrelbrink, J. (2001), "Birch–Tate conjecture", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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