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Period-index bounds for arithmetic threefolds

  • Benjamin AntieauEmail author
  • Asher Auel
  • Colin Ingalls
  • Daniel Krashen
  • Max Lieblich
Article
  • 73 Downloads

Abstract

The standard period-index conjecture for Brauer groups of p-adic surfaces S predicts that \({{\,\mathrm{ind}\,}}(\alpha )|{{\,\mathrm{per}\,}}(\alpha )^3\) for every \(\alpha \in {{\,\mathrm{Br}\,}}(\mathbf {Q}_p(S))\). Using Gabber’s theory of prime-to-\(\ell \) alterations and the deformation theory of twisted sheaves, we prove that \({{\,\mathrm{ind}\,}}(\alpha )|{{\,\mathrm{per}\,}}(\alpha )^4\) for \(\alpha \) of period prime to 6p, giving the first uniform period-index bounds over such fields.

Mathematics Subject Classification

14F22 14J20 16K50 

Notes

Acknowledgements

We would like to thank the Structured Quartet Research Ensembles (SQuaREs) program of the American Institute of Mathematics (AIM) for its hospitality and support for this project. We also thank the Banff International Research Station (BIRS) and the Institute for Computational and Experimental Research in Mathematics (ICERM) for wonderful working environments during the final stages of preparation of this paper. We thank François Charles, Jean-Louis Colliot-Thélène, Ronen Mukamel, R. Parimala, Eryn Schultz, Lenny Taelman, and Tony Várilly-Alvarado for helpful discussions. We especially thank Dan Abramovich, Sam Payne, and Dhruv Ranganathan for expert advice on toroidal geometry. Finally, we are very grateful to Minseon Shin for several comments and corrections on an earlier draft and to the referees for their detailed comments on this paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Benjamin Antieau
    • 1
    Email author
  • Asher Auel
    • 2
  • Colin Ingalls
    • 3
  • Daniel Krashen
    • 4
  • Max Lieblich
    • 5
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicago, IllinoisUSA
  2. 2.Department of MathematicsYale UniversityNew Haven, ConnecticutUSA
  3. 3.School of Mathematics and StatisticsCarlton UniversityOttawa, OntarioCanada
  4. 4.Department of MathematicsRutgers, The State University of New JerseyNewark, New JerseyUSA
  5. 5.Department of MathematicsUniversity of WashingtonSeattle, WashingtonUSA

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