Lichtenbaum's conjecture on special values

Builds on K-groups of rings of integers Dedekind zeta function Algebraic number field

Statement

Let $F$ be a totally real number field, that is, every embedding $F \to \C$ has image in $\R$ , and let $\mathcal{O}_F$ be its ring of integers. Let $\zeta_F$ be the Dedekind zeta function of $F$ , and let $n \ge 2$ be even. Then the K-groups $K_{2n-2}(\mathcal{O}_F)$ and $K_{2n-1}(\mathcal{O}_F)$ are finite, and $\zeta_F(1 - n) = \pm \, 2^{e} \, \frac{\abs{K_{2n-2}(\mathcal{O}_F)}}{\abs{K_{2n-1}(\mathcal{O}_F)}}$ for some integer $e$ .

Remarks

The finiteness is known: it follows from Quillen's finite generation theorem and Borel's computation of ranks, since $r_2 = 0$ , $2n - 2$ is even and $2n - 1 \equiv 3 \pmod 4$ . The value $\zeta_F(1 - n)$ is a nonzero rational number by the Siegel–Klingen theorem. Lichtenbaum also proposed a form for all number fields, in which the leading coefficient of $\zeta_F$ at $1 - n$ is compared with the orders of K-groups and the Borel regulator.

In the form above, which ignores powers of $2$ , the conjecture is now a theorem. It follows from the main conjecture of Iwasawa theory for totally real fields, proved by Wiles, with the norm residue isomorphism theorem of Rost and Voevodsky. The case $n = 2$ is the Birch–Tate conjecture.

Example: for $F = \Q$ and $n = 2$ , $\zeta(-1) = -1/12$ , $\abs{K_2(\Z)} = 2$ and $\abs{K_3(\Z)} = 48$ , and indeed $-1/12 = -2/24 = -2^{1} \cdot 2/48$ .

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Algebraic number field concept
Dedekind zeta function concept
K-groups of rings of integers concept
Lichtenbaum's conjecture on special values conjecture

Tag	`33AYS9Q2PB`
URI	`m2://33AYS9Q2PB`
Kind	conjecture
Writing	W0 stub
MSC	11R70, 11R42, 19F27
Status	This copy of the site has no build data. Run `m2 status 33AYS9Q2PB`.
About	K-groups of rings of integers, Dedekind zeta function
Uses	K-groups of rings of integers, Dedekind zeta function, Algebraic number field
Changed	3 November 2026

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