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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.
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)