On the values of Artin \(L\)-functions. (English) Zbl 1169.11050

Gross, Benedict H.

Pure Appl. Math. Q. 1, No. 1, 1-13 (2005).

This paper was written in 1979 and circulated as a preprint for 25 years. For reasons of historical interest the author decided to publish it in its original form. Let \(f\geq 1\) be an integer, \(\mu_f\) the group of \(f\)-th roots of unity, \(F={\mathbb Q}(\mu_f)\) and \(A={\mathbb Z}[\mu_f]\). There is a canonical isomorphism \[ ({\mathbb Z}/f)^*\rightarrow \text{Gal}(F/{\mathbb Q}),\qquad b\mapsto \text{Frob}(b), \] where \(\zeta^{\text{Frob}(b)}=\zeta^b\). Fix an embedding \(\phi:F\rightarrow {\mathbb C}\). This induces a map \(\phi_*:K_3A\rightarrow K_3{\mathbb C}\) by functoriality. Bloch and Thurston proved that for \(a\in ({\mathbb Z}/f)^*\) there exist elements \(C_f(a)\) in \(K_3A\otimes {\mathbb Q}\) which satisfy \[ C_f(a)^{\text{Frob}(b)}=C_f(ab),\tag{1} \]
\[ C_f(-a)=-C_f(a),\tag{2} \]
\[ e_3(\phi_*C_f(a))=\frac12 \text{dilog}(e^{\frac{2\pi i}{f}}).\tag{3} \] The author proves that for any Dirichlet character \(\chi\) modulo \(f\) the element \(C_\chi=\sum_{({\mathbb Z}/f)^*}\chi(a)\otimes C_f(a)\) spans the \(\chi^{-1}\)-eigenspace of \({\mathbb C}\otimes K_3A\), and the elements \(\{C_f(a): a\in ({\mathbb Z}/f)^*\}\) span the vector space \(K_3A\otimes {\mathbb Q}\). Any relation between them is a consequence of the relation (2) above.

Reviewer: Florin Nicolae (Berlin)

MSC:

11R42	Zeta functions and \(L\)-functions of number fields
11R70	\(K\)-theory of global fields
19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)

Keywords:

values of \(L\)-functions; \(K\)-theory