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Euler characteristic and cohomology of \(\textrm{Sp}_4(\mathbb {Z})\) with nontrivial coefficients

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Abstract

The cohomology of the arithmetic group \(\textrm{Sp}_4(\mathbb {Z})\) with coefficients in any highest weight irreducible representation has been studied. Euler characteristic with coefficients in has been carried out in detail. Combining the results obtained on Euler characteristic and the work of Harder on Eisenstein cohomology (Harder in Geometry and Arithmetic. EMS Series of Congress Reports, European Mathematical Society, Zürich, 2012), the description of the cuspidal cohomology has been achieved. At the end, we employ our study to compute the dimensions for the cohomology spaces .

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Acknowledgements

The authors would like to thank the Max Planck Institute for Mathematics (MPIM), Bonn, for its hospitality where the idea to pursue this project has been initiated during their visit in December, 2016. The authors would also like to thank the referee for carefully reading the paper and giving detailed suggestions for improving it. JB would like to thank the Mathematics Department of the Georg-August University Göttingen for the support, and especially to Valentin Blomer, Harald Helfgott and Thomas Schick for their encouragement during his postdoctoral studies. JB is supported by ERC Consolidator Grants 648329 (codename GRANT, with H. Helfgott as PI) and 681207 (codename GrDyAP, with A. Thom as PI). IH would like to thank Yves Martin for the long discussions on Siegel modular forms during his visit at the CUNY Graduate Center in Spring 2016. MM would like to thank LAMA—Université Paris-Est Marne-la-Vallée for the support and hospitality. MM also wants to thank Institut des Hautes Etudes Scientifiques and Institut Galilée—Université Paris 13 for their hospitality, as part of this work took place during the stay of the third author in these institutions. MM is also thankful to Michael Harris, Nicolas Andruskiewitsch and Roberto Miatello for their support and encouragement. Last but not the least, authors would like to extend their thanks to Günter Harder for many inspiring discussions on the subject and his support during the writing on this article.

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Author notes

  1. Ivan Horozov

    Present address: BCC, City University of New York, 2155 University Ave, Bronx, NY, 10453, USA

  2. Ivan Horozov

    Present address: Department of Mathematics, Columbia University, Room 509, MC 4406, 2990 Broadway, New York, NY, 10027, USA

Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada

    Jitendra Bajpai

  2. Graduate Center, City University of New York, 365 5th Ave, New York, NY, 10016, USA

    Ivan Horozov

  3. LAMA - Université Paris-Est Marne-la-Vallée, 77454, Marne-la-Vallée, France

    Matias Moya Giusti

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  1. Jitendra Bajpai
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  2. Ivan Horozov
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  3. Matias Moya Giusti
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Correspondence to Ivan Horozov.

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Bajpai, J., Horozov, I. & Moya Giusti, M. Euler characteristic and cohomology of \(\textrm{Sp}_4(\mathbb {Z})\) with nontrivial coefficients. European Journal of Mathematics 9, 47 (2023). https://doi.org/10.1007/s40879-023-00630-3

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  • DOI: https://doi.org/10.1007/s40879-023-00630-3

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