Dilogarithm and K3

Remarks

Part 1 is due to Suslin, who related $\mathcal{B}(F)$ to the indecomposable part of $K_3(F)$ for every infinite field $F$; for a number field the decomposable part is torsion. Part 2 holds because $D$ composed with this identification is the Borel regulator on $K_3$, up to a nonzero factor, and Borel's theorem says that its image is a lattice. Part 3 is Zagier's theorem, which follows from Borel's theorem relating the covolume of this lattice to $\zeta_F(2)$. Geometrically, $D(z)$ is the hyperbolic volume of an ideal tetrahedron with cross-ratio $z$, and the Borel class in degree $3$ is the volume class of hyperbolic space.

Example: for $F = \Q(\sqrt{-3})$, $r_1 = 0$, $r_2 = 1$ and $d_F = -3$. Let $z = e^{i\pi/3}$, a root of unity in $F$; then $1 - z = z^{-1}$, so $\delta([z]) = z \wedge z^{-1} = 0$ and $[z]$ lies in $\mathcal{B}(F)$. Here $\zeta_F(2) = \frac{2\pi^2}{9\sqrt{3}} D(z)$, with $D(z) \approx 1.0149$.

Going deeper

For a student

The result is the first nontrivial case of Borel's regulator theorem made explicit with classical functions. It also links number theory with hyperbolic geometry, because $D$ computes volumes of hyperbolic tetrahedra, and hyperbolic $3$-manifolds give elements of Bloch groups. Zagier's conjecture on polylogarithms predicts the analogous statement for $\zeta_F(m)$ with $m \ge 3$ and higher polylogarithms.