Residues on buildings and de Rham cohomology of \(p\)-adic symmetric domains. (English) Zbl 1103.14010

de Shalit, Ehud

Duke Math. J. 106, No. 1, 123-191 (2001).

For a finite extension \(K\) of the field \(\mathbb{Q}_p\), Drinfeld’s \(p\)-adic symmetric domain of dimension \(d\) over \(K\) is defined as the complement \({\mathcal X}\) in \(\mathbb{P}^d\) of the union of all the \(K\)-rational hyperplanes. Endowed with the natural structure of a rigid analytic space, this domain \({\mathcal X}\) is acted on by the group \(G:=\text{PGL}_{d+1}(K)\), and its central role in both the study of Shimura varieties and the representation theory of \(G\) has become manifest during the past three decades. However, in contrast to the classical real symmetric domains, \(p\)-adic symmetric domains are not simply connected in the étale topology, which makes their cohomology theories difficult to tackle. In a fundamental paper published in 1991, P. Schneider and U. Stuhler [Invent. Math. 105, 47–122 (1991; Zbl 0751.14016)] showed how to compute the cohomology of \({\mathcal X}\) for any cohomology theory satisfying certain additional axioms, including the rigid de Rham cohomology. Their approach established, for each \(0\leq k\leq d\), an isomorphism between \(H^k_{\text{dR}}({\mathcal X})\) and a certain space \(C^k\) of \(k\)-cochains on the Bruhat-Tits building \({\mathcal T}\) of \(G\). In the paper under review, the author re-examines this cohomological result by Schneider and Stuhler, together with a number of questions left open back then. In fact, the author developes a considerably modified, somewhat more explicit and combinatorial approach to computing the de Rham cohomology of \(p\)-adic symmetric domains. To this end, he first analyzes local systems on Bruhat-Tits buildings attached to hyperplane arrangements, describes the related harmonic analysis and the corresponding simplicial cohomology, and then uses this explicit description to develop a novel theory of residues on Bruhat-Tits buildings in arbitrary dimension. The notion of residue of a closed form along simplices in a Bruhat-Tits building, combined with a detailed combinatorial study of the ring of associated logarithmic classes, finally leads to a new interpretation of the cohomology of a \(p\)-adic symmetric domain \({\mathcal X}\), which not only recovers the main theorem of Schneider and Stuhler, but also answers (in the affirmative) some of the open questions originating from their work. More precisely, the author’s main theorem states that his so-called residue cochain map establishes an isomorphism of \(G\)-modules \(H^k_{\text{dR}} ({\mathcal X})\cong C^k_{\text{har}}\), where \(C^k_{\text{har}}\) denotes the space of all harmonic \(k\)-cochains on the related Bruhat-Tits building \({\mathcal T}\). In this context, the space \(C^k\) in the isomorphism theorem of Schneider and Stuhler is then recovered by restricting the author’s harmonic \(k\)-cochains to cells of minimal type. The present very detailed paper is enhanced by a fairly extensive appendix on basic facts (with full proofs) concerning rigid de Rham cohomology theory, which is very useful purely as such.

Reviewer: Werner Kleinert (Berlin)

MSC:

14F40	de Rham cohomology and algebraic geometry
20G25	Linear algebraic groups over local fields and their integers
20E42	Groups with a \(BN\)-pair; buildings
14G22	Rigid analytic geometry

Keywords:

rigid geometry; representations; Shimura varieties; Drinfeld modules

Citations:

Zbl 0751.14016

References:

[1]	P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide , Invent. Math. 128 (1997), 329–377. · Zbl 0908.14005 · doi:10.1007/s002220050143
[2]	S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry , Grundlehren Math. Wiss. 261 , Springer, Berlin, 1984. · Zbl 0539.14017
[3]	F. Bruhat and J. Tits, Groupes réductifs sur un corps local , Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–252. · Zbl 0254.14017 · doi:10.1007/BF02715544
[4]	K. Brown, Buildings , Springer, Berlin, 1989.
[5]	D. Bump, Automorphic Forms and Representations , Cambridge Stud. Adv. Math. 55 , Cambridge Univ. Press, Cambridge, 1997. · Zbl 0911.11022
[6]	V. G. Drinfeld, Elliptic modules , Math. USSR-Sb. 23 (1974), 561–592.
[7]	H. Garland, \(p\)-adic curvature and the cohomology of discrete subgroups of \(p\)-adic groups , Ann. of Math. (2) 97 (1973), 375–.\hs423. JSTOR: · Zbl 0262.22010 · doi:10.2307/1970829
[8]	P. Garrett, Buildings and Classical Groups , Chapman & Hall, London, 1997. · Zbl 0933.20019
[9]	E. Groß e-Klönne, de-Rham Kohomologie in der rigiden Analysis , Ph.D. thesis, Münster Univ., Münster, 1998.
[10]	A. Iovita and M. Spiess, Logarithmic differential forms on \(p\)-adic symmetric spaces , preprint, 2000. · Zbl 1100.14505 · doi:10.1215/S0012-7094-01-11023-5
[11]	R. Kiehl, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie , Invent. Math. 2 (1967), 256–273. · Zbl 0202.20201 · doi:10.1007/BF01425404
[12]	P. Monsky, Formal cohomology, II: The cohomology sequence of a pair , Ann. of Math. (2) 88 (1968), 218–238. JSTOR: · Zbl 0162.52601 · doi:10.2307/1970572
[13]	P. Monsky and G. Washnitzer, Formal cohomology, I , Ann. of Math. (2) 88 (1968), 181–217. JSTOR: · Zbl 0162.52504 · doi:10.2307/1970571
[14]	G. A. Mustafin, Non-Archimedean uniformization , Math. USSR-Sb. 34 (1978), 187–214. · Zbl 0411.14006 · doi:10.1070/SM1978v034n02ABEH001156
[15]	P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes , Invent. Math. 56 (1980), 167–189. · Zbl 0432.14016 · doi:10.1007/BF01392549
[16]	P. Orlik and H. Terao, Arrangements of Hyperplanes , Grundlehren Math. Wiss. 300 , Springer, Berlin, 1992. · Zbl 0757.55001
[17]	P. Schneider, The cohomology of local systems on \(p\)-adically uniformized varieties , Math. Ann. 293 (1992), 623–.\hs650. · Zbl 0774.14022 · doi:10.1007/BF01444738
[18]	P. Schneider and U. Stuhler, The cohomology of \(p\)-adic symmetric spaces , Invent. Math. 105 (1991), 47–122. · Zbl 0751.14016 · doi:10.1007/BF01232257
[19]	–. –. –. –., Resolutions for smooth representations of the general linear group over a local field , J. Reine Angew. Math. 436 (1993), 19–32. · Zbl 0780.20026
[20]	P. Schneider and J. Teitelbaum, An integral transform for \(p\)-adic symmetric spaces , Duke Math. J. 86 (1997), 391–.\hs433. · Zbl 0885.14012 · doi:10.1215/S0012-7094-97-08612-9
[21]	J.-P. Serre, Trees , Springer, Berlin, 1980.