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 {\displaystyle K_{2}} group of a field to its Dedekind zeta function.
Statement Formally, the group K 2 {\displaystyle K_{2}} of a number field F {\displaystyle F} is defined as the center of the Steinberg group of the ring of integers of F {\displaystyle F} . K 2 {\displaystyle K_{2}} is also known as the tame kernel of F {\displaystyle F} . The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function ζ F {\displaystyle \zeta _{F}} . More specifically, let F {\displaystyle F} be a totally real number field and let N {\displaystyle N} be the largest positive integer such that the extension of F {\displaystyle F} by the N {\displaystyle N} -th root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that
# K 2 = | N ⋅ ζ F ( − 1 ) | . {\displaystyle \#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