Classical and elliptic polylogarithms and special values of \(L\)-series. (English) Zbl 0990.11041

Zagier, Don; Gangl, Herbert

Gordon, B. Brent (ed.) et al., The arithmetic and geometry of algebraic cycles. Proceedings of the NATO Advanced Study Institute, Banff, Canada, June 7-19, 1998. Vol. 1. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 548, 561-615 (2000).

From the editor’s preface: The first author gave a series of three lectures at the Banff Meeting about classical and elliptic polylogarithms and special values of \(L\)-series. The paper included here was put into a T_EX format by the second author based on Zagier’s three lectures at the meeting.
For an arbitrary number field \(F\), the Dirichlet class number formula expresses the residue at \(s=1\) of the Dedekind zeta function \(\zeta_F(s)\) as the product of a simple factor (involving the class number of the field) with the determinant of a matrix whose entries are logarithms of units in the field. When \(F\) is a totally real number of degree \(n\), a famous theorem by Klingen and Siegel says that the value \(\zeta_F(m)\) for every positive even integer \(m\) is a rational multiple of \(\pi^{mn}\). Zagier has given a conjectural generalization of these two results. Zagier’s conjecture asserts that special values \(\zeta_F(m)\) for arbitrary number fields \(F\) and positive integers \(m\) can be described in terms of special values of a transcendental function depending only on \(m\), that is, the \(m\)th classical polylogarithm function. More generally, Zagier’s conjecture asserts that a special value of an \(L\)-series of “motivic origin” is expressed in terms of some transcendental function.
This paper gives a survey on the present status of the Zagier conjecture and some theoretical and numerical evidence which supports it. Part I gives a review on the polylogarithm conjecture and surveys some known results, in a unified formulation in terms of algebraic \(K\)-theory. In Part II, partial zeta functions \(\zeta_{F,{\mathcal A}}(s)\) of an imaginary quadratic field \(F\) are considered. This leads to a natural conjecture that the partial zeta-values \(D_F^{m-1/2} \zeta_{F,{\mathcal A}}(m)\) should be lifted polylog values. Finally, in Part III, a conjecture which expresses special values of \(L\)-series associated to elliptic curves in terms of “elliptic logarithm functions” is stated. Again, algebraic \(K\)-theory presents a natural way of understanding the conjectures.
For the entire collection see [Zbl 0933.00032].

MSC:

11G55	Polylogarithms and relations with \(K\)-theory
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

elliptic polylogarithms; special values of \(L\)-series; Dedekind zeta function; Zagier conjecture; partial zeta functions; imaginary quadratic field