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On the $$\lambda$$-adic representations associated to some simple Shimura varieties. (English) Zbl 0765.22011
This paper is a contribution to the programme of associating $$\lambda$$- adic Galois representations to “cohomological” automorphic representations. The author points out that when one has a $$CM$$ field $$F$$ and a division algebra $$D$$ defined and central over $$F$$ then the Shimura varieties which are ‘moduli spaces for abelian varieties with complex multiplication by $$D$$’ are particularly susceptible to analysis. He proves a very sharp theorem of the type mentioned above but it would take too much preparation to repeat the formulation here. The proof uses a “pseudo-stabilized” trace formula and a comparison with the Lefschetz formula. It is the consequence of a series of papers published by the author and this paper relies heavily on its predecessors.

##### MSC:
 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 14G35 Modular and Shimura varieties
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