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Invariant Brauer group of an abelian variety

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Abstract

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper bound on the rank. We exhibit many cases in which the invariant Brauer group is zero, and construct complex abelian varieties in every dimension starting with 2, both simple and non-simple, with invariant Brauer group of order 2. We also address the situation in finite characteristic and over non-closed fields.

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Acknowledgements

The fourth named author (Y.Z.) was partially supported by Simons Foundation Collaboration grant #585711. Part of this work was done during his stay in December 2019 and January 2020 at the Weizmann Institute of Science, Department of Mathematics, whose hospitality and support are gratefully acknowledged.

The authors are grateful to the referee for the careful reading of the paper and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Manchester, Alan Turing Building, Oxford Road, Manchester, M13 9PL, England, UK

    Martin Orr

  2. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2BZ, England, UK

    Alexei N. Skorobogatov

  3. Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow, 127994, Russia

    Alexei N. Skorobogatov

  4. Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Domenico Valloni

  5. Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania, 16802, USA

    Yuri G. Zarhin

Authors
  1. Martin Orr
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  2. Alexei N. Skorobogatov
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  3. Domenico Valloni
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  4. Yuri G. Zarhin
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Correspondence to Alexei N. Skorobogatov.

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Orr, M., Skorobogatov, A.N., Valloni, D. et al. Invariant Brauer group of an abelian variety. Isr. J. Math. 249, 695–733 (2022). https://doi.org/10.1007/s11856-022-2323-5

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  • DOI: https://doi.org/10.1007/s11856-022-2323-5

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