## The trace formula and Drinfeld’s upper halfplane.(English)Zbl 0855.11061

$$K$$ is a $$p$$-adic local field; $$d$$ is an integer $$\geq 2$$; $$D$$ is the division algebra over $$K$$ with invariant $$1/d$$; $$D^*$$ is the multiplicative group of $$D$$; $$\Omega \subset \mathbb{P}_K^{d-1}$$ is the rigid space obtained from $$\mathbb{P}^{d-1}$$ by deleting all $$K$$-rational hyperplanes. Drinfeld’s symmetric space $$\Omega$$ parametrizes a certain type of formal groups. $$\Omega$$ has étale coverings given by division points of this (universal) formal group. Drinfeld has conjectured that the rigid étale cohomology groups of those coverings of $$\Omega$$ give rise to a Jacquet-Langlands correspondence between irreducible discrete-series representations of $$GL_d (K)$$ and irreducible representations of $$D^*$$. For $$d> 2$$ this was shown by Deligne and Kazdan. For $$d=2$$ there is a sketch of the proof of this conjecture by Carayol.
The contribution of this paper is: completing the arguments of Carayol for $$d=2$$ by developing and using a Lefschetz fixed point formula for symmetric spaces and some of their quotients by discrete groups. For any $$d$$ such a fixed point formula is developed. For $$d=2$$ an elementary proof is given.
For $$d>2$$ the fixed point formula does not give sufficient information. One needs some vanishing theorems for its application.
The paper can be seen as a continuation of the work of P. Schneider and U. Stuhler [Invent. Math. 105, No. 1, 47-122 (1991; Zbl 0751.14016)].

### MSC:

 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F85 $$p$$-adic theory, local fields 14F20 Étale and other Grothendieck topologies and (co)homologies

Zbl 0751.14016
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### References:

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