Borel regulator

Remarks

Complex conjugation acts on $\R^\Sigma$ through $\Sigma$. The image of $r_m$ lies in the subspace on which conjugation acts by the sign $(-1)^{m-1}$. This subspace has dimension $d_m$, where $d_m = r_1 + r_2$ for $m$ odd and $d_m = r_2$ for $m$ even. Borel proved that $r_m$ has finite kernel and that its image is a lattice in this subspace. The covolume $R_m(F)$ of this lattice is the Borel regulator of $F$ in degree $m$. Let $\zeta_F^*(1 - m)$ be the leading coefficient of the Dedekind zeta function at $s = 1 - m$. Borel proved that $\zeta_F^*(1 - m) / R_m(F)$ is a nonzero rational number. Normalisations of $b_m$ vary between authors by rational factors and powers of $2 \pi i$.

Example: the analogue for $m = 1$ is the map $\mathcal{O}_F^\times = K_1(\mathcal{O}_F) \to \R^\Sigma$, $u \mapsto (\log \abs{\sigma(u)})_\sigma$, whose image has covolume equal, up to a simple factor, to the classical regulator.

Non-example: for $F = \Q$ and $m$ even, $d_m = 0$, so $r_m$ is zero; this matches the finiteness of $K_{2m-1}(\Z)$ for even $m$.