Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Ghost classes in the cohomology of the Shimura variety associated to $GSp_4$
HTML articles powered by AMS MathViewer

by Matias Victor Moya Giusti PDF
Proc. Amer. Math. Soc. 146 (2018), 2315-2325 Request permission

Abstract:

In this paper we study the existence of ghost classes in the cohomology of the Shimura variety associated to the group of symplectic similitudes $GSp_4$. The existence of ghost classes for the trivial coefficient system is known. We show that ghost classes only exist for the trivial coefficient system and they lie in the cohomology group in degree 2. Moreover we prove that the weight of the mixed Hodge structure associated to the space of ghost classes is the middle weight.
References
  • A. Borel, Cohomology and spectrum of an arithmetic group, Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 28–45. MR 731761
  • A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. MR 387495, DOI 10.1007/BF02566134
  • Jens Franke, Harmonic analysis in weighted $L_2$-spaces, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 2, 181–279 (English, with English and French summaries). MR 1603257, DOI 10.1016/S0012-9593(98)80015-3
  • Neven Grbac and Harald Grobner, The residual Eisenstein cohomology of $Sp_4$ over a totally real number field, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5199–5235. MR 3074371, DOI 10.1090/S0002-9947-2013-05796-0
  • G. Harder, Eisenstein cohomology of arithmetic groups. The case $\textrm {GL}_2$, Invent. Math. 89 (1987), no. 1, 37–118. MR 892187, DOI 10.1007/BF01404673
  • Günter Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive, Lecture Notes in Mathematics, vol. 1562, Springer-Verlag, Berlin, 1993 (German). MR 1285354, DOI 10.1007/BFb0090305
  • Günter Harder, The Eisenstein motive for the cohomology of $\textrm {GSp}_2(\Bbb Z)$, Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 143–164 (English, with English and German summaries). MR 2987659, DOI 10.4171/119-1/10
  • Michael Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio Math. 60 (1986), no. 3, 323–378. MR 869106
  • Michael Harris and Steven Zucker, Boundary cohomology of Shimura varieties. II. Hodge theory at the boundary, Invent. Math. 116 (1994), no. 1-3, 243–308. MR 1253194, DOI 10.1007/BF01231562
  • M. Harris and S. Zucker, Erratum: “Boundary cohomology of Shimura varieties. II. Hodge theory at the boundary” [Invent. Math. 116 (1994), no. 1-3, 243–308; MR1253194 (95f:14041)], Invent. Math. 121 (1995), no. 2, 437. MR 1346216, DOI 10.1007/BF01884308
  • A. Kewenig and T. Rieband, Geisterklassen im Bild der Borelabbildung fur symplektische und orthogonale Gruppen, Diplomarbeit, Mathematisches Institut der Rheinisch Friedrich-Wilhelms-Universität Bonn, Bonn, 1997 (unpublished).
  • Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
  • J. S. Milne, Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 265–378. MR 2192012
  • Joachim Schwermer, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Mathematics, vol. 988, Springer-Verlag, Berlin, 1983 (German). MR 822473, DOI 10.1007/BFb0070268
  • Joachim Schwermer, Cohomology of arithmetic groups, automorphic forms and $L$-functions, Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989) Lecture Notes in Math., vol. 1447, Springer, Berlin, 1990, pp. 1–29. MR 1082960, DOI 10.1007/BFb0085724
  • Steven Zucker, Locally homogeneous variations of Hodge structure, Enseign. Math. (2) 27 (1981), no. 3-4, 243–276 (1982). MR 659151
  • Steven Zucker, On the boundary cohomology of locally symmetric varieties, Vietnam J. Math. 25 (1997), no. 4, 279–318. MR 1679475
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14G35, 14D07
  • Retrieve articles in all journals with MSC (2010): 14G35, 14D07
Additional Information
  • Matias Victor Moya Giusti
  • Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • Address at time of publication: CIEM-FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n - Ciudad Universitaria, CP. 5016, Córdoba, Argentina
  • Email: moya@famaf.unc.edu.ar
  • Received by editor(s): November 12, 2016
  • Received by editor(s) in revised form: March 21, 2017
  • Published electronically: March 9, 2018
  • Additional Notes: This paper is based on part of the author’s doctoral thesis at Université Paris Diderot - Paris 7, written under the direction of Michael Harris, where the author was supported by the ERC (European Research Council).
  • Communicated by: Romyar T. Sharifi
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 2315-2325
  • MSC (2010): Primary 14G35, 14D07
  • DOI: https://doi.org/10.1090/proc/13788
  • MathSciNet review: 3778137