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

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