Birch–Tate conjecture

In mathematics, specifically in algebraic K-theory, the Birch–Tate conjecture is a conjecture proposed by Bryan John Birch and John Tate relating the $K_{2}$ group of a field to its Dedekind zeta function.

Statement

Formally, the group $K_{2}$ of a number field $F$ is defined as the center of the Steinberg group of the ring of integers of $F$. $K_{2}$ 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 positive integer such that the extension of $F$ by the $N$-th root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that

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

Status

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."

References

• Tate, J. T. (1971). "Symbols in arithmetic". In Berger, M.; Dieudonné, J.; Leray, J.; Lions, J.-L.; Malliavin, P.; Serre, J.-P. (eds.). Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1. Paris: Gauthier-Villars. pp. 201–211.

External links

• Hurrelbrink, J. (2001) [1994], "Birch–Tate conjecture", Encyclopedia of Mathematics, EMS Press