## Signature defects of cusps and values of L-functions: The nonsplit case.(English)Zbl 0577.58030

This note is a supplement to our paper [Ann. Math., II. Ser. 118, 131-177 (1983; Zbl 0531.58048)]. Hirzebruch conjectured that the values at zero of the Shimizu L-functions are realized as the signature defects of cusps associated to Hilbert modular varieties. In the paper cited above we claimed to have established the Hirzebruch conjecture but, as was pointed out to us by W. Müller, we only dealt with the ”split” case. In fact our method of proof extends with essentially no change to the non-split case.

### MSC:

 58J35 Heat and other parabolic equation methods for PDEs on manifolds 57R20 Characteristic classes and numbers in differential topology 53C05 Connections (general theory) 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 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11R80 Totally real fields 14B05 Singularities in algebraic geometry 14G25 Global ground fields in algebraic geometry 14J17 Singularities of surfaces or higher-dimensional varieties 14J25 Special surfaces

### Citations:

Zbl 0285.14007; Zbl 0297.58008; Zbl 0531.58048
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