The trace formula and Drinfeld's upper halfplane
\(K\) 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 \textit{P. Schneider} and \textit{U. Stuhler} [Invent. Math. 105, No. 1, 47-122 (1991; Zbl 0751.14016)].
scientific article; zbMATH DE number 727832
scientific article