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Signature des variétés modulaires de Hilbert et représentations diédrales. (Signature of Hilbert modular varieties and dihedral representations). (French) Zbl 0723.11019
Cohomology of arithmetic groups and automorphic forms, Proc. Conf., Luminy/Fr. 1989, Lect. Notes Math. 1447, 249-260 (1990).
[For the entire collection see Zbl 0706.00009.]
The signature of a Hilbert modular variety $$\Gamma \setminus G_{\infty}/K_{\infty}$$ $$(G_{\infty}=SL(2,k\otimes_{{\mathbb{Q}}}{\mathbb{R}})$$, k a totally real number field, $$K_{\infty}=$$ maximal compact subgroup, $$\Gamma$$ an arithmetic subgroup) is computed here as a trace on the $$L^ 2$$- cohomology space of degree $$d=complex$$ dimension of the variety. More generally, the trace of $$\sigma\otimes h$$, $$\sigma =$$ the signature operator, h an element of the Hecke algebra is computed. The $$L^ 2$$- cohomology is isomorphic to Lie algebra cohomology with values in $$L^ 2(\Gamma \setminus G_{\infty})$$. Using the spectral decomposition of $$L^ 2(\Gamma \setminus G_{\infty})$$ and pseudo-coefficients for the discrete series representations, the signature is expressed as the trace of an operator in the discrete part of $$L^ 2(SL(2,k)\setminus SL(2,{\mathbb{A}}))$$, to which the trace formula is applied. This gives the index formula for the Hecke equivariant signature.
It is shown that only dihedral automorphic representations give a contribution to the signature.
##### MSC:
 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; trace formulas (e.g., that of Selberg)