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

Atiyah, Michael F.; Donnelly, H.; Singer, I. M.

Ann. Math. (2) 119, 635-637 (1984).

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

Keywords:

eta invariants; signature defects of cusps; special values of; L- functions; cusp on Hilbert modular variety; lattice in totally; real field; Hirzebruch L-polynomial; Hirzebruch; signature theorem; flat connection; Feynman-Kac; representation of the heat kernel

Citations:

Zbl 0285.14007; Zbl 0297.58008; Zbl 0531.58048