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Topological polylogarithms and \(p\)-adic interpolation of \(L\)-values of totally real fields. (English) Zbl 1448.11122
Summary: We develop the topological polylogarithm which provides an integral version of Nori’s Eisenstein cohomology classes for \({{\mathrm{GL}}}_n(\mathbb {Z})\) and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of \(L\)-functions of totally real fields and a construction of the associated \(p\)-adic \(L\)-function. Using a result of P. Graf [“Polylogarithms for \(\mathrm{GL}_2\) over totally real fields”, Preprint, arXiv:1604.04209], we also apply this to prove some integrality and \(p\)-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.

MSC:
11G55 Polylogarithms and relations with \(K\)-theory
11R42 Zeta functions and \(L\)-functions of number fields
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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