×

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

Citations:

Zbl 0751.14016
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. G. Berkovich, Étale cohomology for non-Archimedean analytic spaces , Inst. Hautes Études Sci. Publ. Math. (1993), no. 78, 5-161 (1994). · Zbl 0804.32019
[2] V. Berkovich, Vanishing cycles for formal schemes , Invent. Math. 115 (1994), no. 3, 539-571. · Zbl 0791.14008
[3] J. Bernstein, P. Deligne, K. Kazhdan, and M. F. Vigneras, Représentations des Groupes Réductifs sur un Corps Local , Hermann, Paris, 1984. · Zbl 0544.00007
[4] H. Carayol, Nonabelian Lubin-Tate theory , Automorphic forms, Shimura varieties, and \(L\)-functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 15-39. · Zbl 0704.11049
[5] P. Cartier, Sur les Représentations des Groupes Réductifs \(p\)-adiques et leur Characteres , Lecture Notes in Math., vol. 569, Springer, Berlin. · Zbl 0421.22010
[6] P. Deligne, Séminaire de Géometrie Algébrique du Bois-Marie SGA \(4 1/2\) , Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977. · Zbl 0345.00010
[7] V. Drinfeld, Coverings of \(p\)-adic symmetric domains , Functional Anal. Appl. 10 (1976), 107-115. · Zbl 0346.14010
[8] R. Elkik, Solutions d’équations à coefficients dans un anneau hensélien , Ann. Sci. École Norm. Sup. (4) 6 (1973), 553-603 (1974). · Zbl 0327.14001
[9] G. Faltings, Crystalline cohomology of semistable curves-the \(\mathbbQ_p\)-theory , submitted to J. Algebraic Geom. · Zbl 0883.14007
[10] K. Fujiwara, Rigid geometry, Lefschetz-Verdier trace formula, and Deligne’s conjecture , manuscript, 1992. · Zbl 0920.14005
[11] A. Genestier, Ramification du Revetement de Drinfeld , thesis, Paris-Orsay, 1993. · Zbl 0974.11029
[12] W. Lütkebohmert, Riemann’s existence problem for a \(p\)-adic field , Invent. Math. 111 (1993), no. 2, 309-330. · Zbl 0780.32005
[13] P. Schneider and U. Stuhler, The cohomology of \(p\)-adic symmetric spaces , Invent. Math. 105 (1991), no. 1, 47-122. · Zbl 0751.14016
[14] P. Schneider and U. Stuhler, Cohomologie \(\ell\)-adique et Fonctions \(L\) , Lecture Notes in Math., vol. 589, Seminaire de Geometrie Algebrique 5, Berlin, 1991.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.