Regulators and Special Values of L-Functions

Regulators are maps from algebraic K-theory to real-valued cohomology theories that measure the "transcendental" part of K-groups. The Beilinson conjectures predict that the regulators compute special values of L-functions, providing the deepest known connection between algebra and analysis.

The Borel regulator

Definition6.6Borel regulator

For a number field $F$ with ring of integers $\mathcal{O}_F$ , the Borel regulator is a map

$r_{\text{Borel}}: K_{2n-1}(\mathcal{O}_F) \to \mathbb{R}^{d_n}$

where $d_n = \begin{cases} r_1 + r_2 & n \text{ odd} \\ r_2 & n \text{ even} \end{cases}$ ( $r_1$ = real places, $r_2$ = complex places).

The map is defined via the van Est isomorphism: the inclusion $GL_N(\mathcal{O}_F) \hookrightarrow GL_N(F \otimes_\mathbb{Q} \mathbb{R}) = \prod_\sigma GL_N(\mathbb{R} \text{ or } \mathbb{C})$ induces maps on group cohomology. The Borel regulator is the composition:

$K_{2n-1}(\mathcal{O}_F) \to H_{2n-1}(GL(\mathcal{O}_F); \mathbb{Z}) \to H_{2n-1}(GL(\mathbb{R})^{r_1} \times GL(\mathbb{C})^{r_2}; \mathbb{R}) \to \mathbb{R}^{d_n}$

where the last map uses the continuous cohomology class corresponding to the Borel class $b_{2n-1} \in H^{2n-1}_c(GL(\mathbb{R}); \mathbb{R})$ .

Theorem6.3Borel's theorem

The image of $r_{\text{Borel}}: K_{2n-1}(\mathcal{O}_F) \otimes \mathbb{R} \to \mathbb{R}^{d_n}$ is a lattice of rank $d_n$ . The Borel regulator $R_n(F)$ is the covolume of this lattice:

$R_n(F) = \operatorname{vol}(\mathbb{R}^{d_n} / r_{\text{Borel}}(K_{2n-1}(\mathcal{O}_F)_{\text{free}})).$

This is well-defined up to a rational number. For $n = 1$ : $K_1(\mathcal{O}_F) = \mathcal{O}_F^{\times}$ and $R_1(F)$ is the classical regulator $R_F$ from the Dirichlet unit theorem, up to a rational factor.

The Beilinson conjectures

Definition6.7Beilinson regulator

For a smooth projective variety $X$ over $\mathbb{Q}$ , Beilinson defines a regulator map from motivic cohomology to Deligne cohomology:

$r_\mathcal{D}: H^n_{\text{mot}}(X, \mathbb{Q}(p)) \to H^n_\mathcal{D}(X/\mathbb{R}, \mathbb{R}(p))$

where Deligne cohomology $H^n_\mathcal{D}$ is a real-valued cohomology theory combining Hodge theory and de Rham cohomology:

$H^n_\mathcal{D}(X/\mathbb{R}, \mathbb{R}(p)) = H^n(X(\mathbb{C}), \mathbb{R}(p) \to \Omega^0_X \to \cdots \to \Omega^{p-1}_X)$

(the hypercohomology of the Deligne complex).

For $X = \operatorname{Spec} \mathcal{O}_F$ and the weight- $(2n-1, n)$ part: $r_\mathcal{D}$ coincides with the Borel regulator on $K_{2n-1}(\mathcal{O}_F)$ .

RemarkBeilinson's conjectures

Beilinson's conjecture (1984) predicts, for a smooth projective variety $X/\mathbb{Q}$ and its $L$ -function $L(H^i(X), s)$ :

The order of vanishing of $L(H^i(X), s)$ at $s = m$ (an integer $\leq i/2$ ) equals the rank of the motivic cohomology group $H^{i+1}_{\text{mot}}(X, \mathbb{Q}(i+1-m))$ .
The leading coefficient of $L(H^i(X), s)$ at $s = m$ is given (up to a rational number) by the determinant of the Beilinson regulator matrix:

$L^*(H^i(X), m) \sim_{\mathbb{Q}^{\times}} \det(r_\mathcal{D}: H^{i+1}_{\text{mot}}(X, \mathbb{Q}(i+1-m)) \to H^{i+1}_\mathcal{D}(X/\mathbb{R}, \mathbb{R}(i+1-m))).$

This vastly generalizes the analytic class number formula, Dirichlet's class number formula, and the Birch--Swinnerton-Dyer conjecture.

Known cases

ExampleThe Dirichlet regulator (n = 1)

For $n = 1$ , $X = \operatorname{Spec} \mathcal{O}_F$ : $K_1(\mathcal{O}_F) = \mathcal{O}_F^{\times}$ , and the Beilinson regulator is the logarithmic embedding:

$r_\mathcal{D}: \mathcal{O}_F^{\times} \otimes \mathbb{R} \to \mathbb{R}^{r_1 + r_2 - 1}, \quad u \mapsto (\log|\sigma_i(u)|)_i$

where $\sigma_i$ are the archimedean embeddings. The covolume $R_1(F) = R_F$ is the Dirichlet regulator. The class number formula:

$\lim_{s \to 1} (s-1) \zeta_F(s) = \frac{2^{r_1}(2\pi)^{r_2} R_F h_F}{w_F \sqrt{|d_F|}}$

expresses $\zeta_F^*(1)$ in terms of the regulator, class number $h_F$ , roots of unity $w_F$ , and discriminant $d_F$ . This is the archetype for all Beilinson conjectures.

ExampleBorel regulator for K₃

For $K_3(\mathcal{O}_F)$ : the Borel regulator gives $r_{\text{Borel}}: K_3(\mathcal{O}_F) \otimes \mathbb{R} \to \mathbb{R}^{r_2}$ (only complex places contribute for weight 2).

Zagier's conjecture (proved by Goncharov for $K_3$ ): the Borel regulator on $K_3(F) \cong \mathbb{Z}^{r_2} \oplus (\text{torsion})$ is given by the Bloch--Wigner dilogarithm:

$D(z) = \operatorname{Im}(\operatorname{Li}_2(z)) + \arg(1-z) \log|z|$

applied to elements of the Bloch group $\mathcal{B}(F) \cong K_3^{\text{ind}}(F)$ (the indecomposable $K_3$ ). Specifically, for each complex embedding $\sigma: F \hookrightarrow \mathbb{C}$ :

$r_\sigma([z]_2) = D(\sigma(z))$

where $[z]_2 \in \mathcal{B}(F)$ is the element corresponding to $z \in F \setminus \{0, 1\}$ .

This connects $K_3$ to hyperbolic geometry: $D(z)$ is the volume of an ideal tetrahedron in $\mathbb{H}^3$ with cross-ratio $z$ , and $\zeta_F(2) \sim R_2(F)$ via the Borel regulator.

The Birch--Swinnerton-Dyer connection

RemarkBSD as a special case of Beilinson

For an elliptic curve $E/\mathbb{Q}$ , the Birch--Swinnerton-Dyer conjecture is a special case of Beilinson's conjectures. Taking $X = E$ , $i = 1$ , $m = 1$ :

$\operatorname{ord}_{s=1} L(E, s) = \operatorname{rk} E(\mathbb{Q}) = \operatorname{rk} H^2_{\text{mot}}(E, \mathbb{Z}(1)) = \operatorname{rk} CH^1(E, 1) = \operatorname{rk} \operatorname{Pic}^0(E) \otimes \mathbb{Q}$

Wait, more precisely: the relevant motivic cohomology group is $H^1_{\text{mot}}(E, \mathbb{Q}(1))$ , and the regulator is the Neron--Tate height pairing. The leading coefficient of $L(E, s)$ at $s = 1$ involves:

$L^*(E, 1) \sim \frac{\Omega_E \cdot R_E \cdot \prod c_p \cdot |\operatorname{Sha}|}{|E(\mathbb{Q})_{\text{tors}}|^2}$

where $R_E$ is the regulator (determinant of the height pairing matrix), $\Omega_E$ is the real period, $c_p$ are Tamagawa numbers, and $\operatorname{Sha}$ is the Tate--Shafarevich group (conjectured finite).

The K-theoretic formulation packages all these invariants into a single regulator computation.