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Transgressed Euler classes of \(SL(2n,\mathbb{Z})\) vector bundles, adiabatic limits of eta invariants and special values of \(L\)-functions. (English) Zbl 0768.58048
The Hirzebruch conjecture on the signature of the Hilbert modular varieties was first proved by Atiyah-Donnelly-Singer and Müller. Sullivan showed that the Euler class of an \(SL(2n,\mathbb{Z})\) vector bundle vanishes rationally. This paper is devoted to the study of the deep relation between these results. The paper is divided into five sections. In Section 1 a canonical transgression of the Euler class of a \(GL(2n;\mathbb{Z})\) vector bundle is considered. The interpretation of the results of Section 1 in the context of the local families index theorem is in Section 2. Section 3 is devoted to the calculation of the adiabatic limit of the eta invariant of the signature operator. The purpose of Section 4 is to specialize the results of Section 3 to torus fibrations over homogeneous vector bundles. The purpose of Section 5 is to analyze more details of some properties of the eta invariants from Section 3.

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
57R20 Characteristic classes and numbers in differential topology
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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