Birch–Tate conjecture

Problem's Description

The conjecture known as the Birch–Tate conjecture within the field of algebraic K-theory, and was jointly proposed by Bryan John Birch and John Tate.

Let $K$ represent a totally real number field, $\zeta_K$ denote the Dedekind zeta function, and $K_2(\mathcal{O}_K)$ be the algebraic $K_2$-group.

The Birch-Tate conjecture suggests that the equation $$\zeta_K(-1) = \left| \frac{{K_2(\mathcal{O}_K)}}{{H^0(K, \mathbb{Q}/\mathbb{Z}(2))}} \right|$$ holds, potentially with a power of 2 adjustment.

This conjecture emerged following John Tate's establishment of a similar equation for function fields, which heavily drew from André Weil's resolution of the Riemann hypothesis for curves.

We refer to [1] for this conjecture.

1. ChapterIs an originSymbols in arithmetic

   • John Tate

   pp. 201-211, year of publication: 1971