×

zbMATH — the first resource for mathematics

Control theorems of \(p\)-nearly ordinary cohomology groups for \(\text{SL} (n)\). (English) Zbl 0852.11023
Let \(p\) be a prime. The author proves control theorems (with finite error: Theorems 5.1, 5.2, 6.1; exact control theorem for Borel subgroups: Theorem 7.1) and independence of weight (Theorem 4.1) for the \(p\)-adic nearly ordinary cohomology groups for \(\text{SL} (n)\) over an arbitrary number field. All these generalize the results already obtained for SL(2) [the author, \(p\)-ordinary cohomology groups for SL(2) over number fields, Duke Math. J. 69, 259-314 (1993)].
Possible implications in the study of \(p\)-adic cohomological modular forms on \(\text{GL} (n)\) and finiteness theorems for the corresponding \(p\)-adic nearly ordinary Hecke algebras are indicated.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F75 Cohomology of arithmetic groups
11F85 \(p\)-adic theory, local fields
PDF BibTeX XML Cite
Full Text: DOI Link Numdam EuDML
References:
[1] ARTHUR (J.) and CLOZEL (L.) . - Simple algebras, base change, and the advanced theory of the trace formula , Ann. of Math. Studies, 120, Princeton University Press, 1989 . MR 90m:22041 | Zbl 0682.10022 · Zbl 0682.10022
[2] BOREL (A.) . - Regularization theorems in Lie algebra cohomology. Applications , Duke Math. J., t. 50, 1983 , p. 605-623. Article | MR 84h:17009 | Zbl 0528.22010 · Zbl 0528.22010 · doi:10.1215/S0012-7094-83-05028-7 · minidml.mathdoc.fr
[3] BOREL (A.) and SERRE (J.-P.) . - Corners and arithmetic groups , Comment. Math. Helv., t. 48, 1974 , p. 436-491. MR 52 #8337 | Zbl 0274.22011 · Zbl 0274.22011 · doi:10.1007/BF02566134 · eudml:139559
[4] BOREL (A.) and WALLACH (N.) . - Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups , Ann. of Math. Studies 94, Princeton University Press, Princeton, 1980 . MR 83c:22018 | Zbl 0443.22010 · Zbl 0443.22010
[5] CLOZEL (L.) . - Motifs et formes automorphes : applications du principe de fonctorialité , Perspective Math., t. 10, 1990 , p. 77-159. MR 91k:11042 | Zbl 0705.11029 · Zbl 0705.11029
[6] CLOZEL (L.) . - Représentations galoisienne associées aux représentations automorphes autoduales de GL(n) , Publ. I.H.E.S., t. 73, 1991 , p. 97-145. Numdam | MR 92i:11055 | Zbl 0739.11020 · Zbl 0739.11020 · doi:10.1007/BF02699257 · numdam:PMIHES_1991__73__97_0 · eudml:104076
[7] CLOZEL (L.) . - On the cohomology of Kottwitz’s arithmetic varieties , Duke Math. J., t. 72, 1993 , p. 757-795. Article | MR 95b:11055 | Zbl 0974.11019 · Zbl 0974.11019 · doi:10.1215/S0012-7094-93-07229-8 · minidml.mathdoc.fr
[8] DELIGNE (P.) , KAZHDAN (D.) and VIGNÉRAS (M.-F.) . - Représentations des algèbres centrales simples p-adiques, in ”Représentations des groupes réductifs sur un corps local” , Hermann, Paris, 1984 , p. 33-117. MR 86h:11044 | Zbl 0583.22009 · Zbl 0583.22009
[9] HIDA (H.) . - On p-adic Hecke algebras for GL2 over totally real fields , Ann. of Math., t. 128, 1988 , p. 295-384. MR 89m:11046 | Zbl 0658.10034 · Zbl 0658.10034 · doi:10.2307/1971444
[10] HIDA (H.) . - p-Ordinary cohomology groups for SL(2) over number fields , Duke Math. J., t. 69, 1993 , p. 259-314. Article | MR 94g:11031 | Zbl 0941.11024 · Zbl 0941.11024 · doi:10.1215/S0012-7094-93-06914-1 · minidml.mathdoc.fr
[11] HIDA (H.) . - Modular p-adic L-functions and p-adic Hecke algebras , in Japanese, Sugaku, t. 44, n^\circ 4, 1992 , p. 289-305. (English transl. : Amer. Math. Soc. Transl. 160 (2), 1994 , p. 125-154.). Zbl 0811.11040 · Zbl 0811.11040
[12] HIDA (H.) . - p-Adic ordinary Hecke algebras for GL(2) , Ann. Inst. Fourier 44, 1994 , p. 1289-1323. Numdam | MR 95k:11065 | Zbl 0819.11017 · Zbl 0819.11017 · doi:10.5802/aif.1434 · numdam:AIF_1994__44_5_1289_0 · eudml:75098
[13] JANTZEN (J.C.) . - Representations of algebraic groups , Pure and Applied Math. 131, Academic Press, 1987 . MR 89c:20001 | Zbl 0654.20039 · Zbl 0654.20039
[14] MATSUSHIMA (Y.) and MURAKAMI (S.) . - On vector bundle valued harmonic forms and automorphic forms on Symmetric Riemannian manifolds , Ann. of Math., t. 78, 1963 , p. 365-416. MR 27 #2997 | Zbl 0125.10702 · Zbl 0125.10702 · doi:10.2307/1970348
[15] MOEGLIN (C.) and WALDSPURGER (J.-L.) . - Le spectre résiduel de GL(n) , Ann. Scient. École Norm. Sup., 4e série, t. 22, 1989 , p. 605-674. Numdam | MR 91b:22028 | Zbl 0696.10023 · Zbl 0696.10023 · numdam:ASENS_1989_4_22_4_605_0 · eudml:82264
[16] ROGAWSKI (J.D.) . - Representations of GL(n) and division algebras over a p-adic fields , Duke Math. J., t. 50, 1983 , p. 161-196. Article | MR 84j:12018 | Zbl 0523.22015 · Zbl 0523.22015 · doi:10.1215/S0012-7094-83-05006-8 · minidml.mathdoc.fr
[17] WILSON (B.) . - p-Adic Hecke algebra, Thesis , UCLA, 1993 .
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.