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.


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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