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Degeneration of polylogarithms and special values of \(L\)-functions for totally real fields. (English) Zbl 1183.11024

A conjecture of Zagier describes the values of \(L\)-functions of a number fields in terms of polylogs. The present paper uses the functoriality of the topological polylog on the universal abelian scheme over a Hilbert modular variety to describe the degeneration of certain Eisenstein classes in terms of special values of partial \(L\)-functions. The problem is reduced to explicit computations of Nori and Sczech, which are reproduced in the paper. The main result was also proved by Blottière, using different methods

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11G55 Polylogarithms and relations with \(K\)-theory
11R42 Zeta functions and \(L\)-functions of number fields
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