Goncharov, Alexander B. Polylogarithms in arithmetic and geometry. (English) Zbl 0849.11087 Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 374-387 (1995). Having recalled the definition and the main properties of the dilogarithm, the author outlines its role in the \(K\)-theory of fields and in hyperbolic geometry. He then writes: “In this lecture I will explain how most of these facts about the dilogarithm are generalized to the trilogarithm and outline what should happen in general.” We list the main topics discussed by the author following his table of contents: the trilogarithm and the value of the Dedekind function \(\zeta_F (s)\) of a number field \(F\) at \(s=3\); the trilogarithm and algebraic \(K\)-theory; classical polylogarithms and motivic complexes; Zagier’s conjecture; motivic complexes for curves; explicit formulae for regulators in the case of curves; special values of \(L\)-functions of elliptic curves; motivic Lie algebra, its enveloping algebra, and framed mixed Tate motives; hyperbolic geometry; hyperlogarithms; the quantum dilogarithm of Faddeev and Kashaev. It is a concise and relatively self-contained exposition of the results and conjectures discussed in detail in a series of the author’s recent papers.For the entire collection see [Zbl 0829.00014]. Reviewer: B.Z.Moroz (Bonn) Cited in 1 ReviewCited in 36 Documents 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) 14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Keywords:dilogarithm; \(K\)-theory of fields; hyperbolic geometry; Dedekind function; polylogarithms; motivic complexes; Zagier’s conjecture; curves; regulators; special values of \(L\)-functions; motivic Lie algebra; framed mixed Tate motives; hyperlogarithms × Cite Format Result Cite Review PDF