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.
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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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Ben Antieau was partially supported by NSF CAREER Grant DMS-1552766 and NSF Grants DMS-1358832 and DMS-1461847. Asher Auel was partially supported by NSA Young Investigator Grants H98230-13-1-0291 and H98230-16-1-0321. Colin Ingalls was partially supported by an NSERC Discovery grant. Daniel Krashen was partially supported by NSF CAREER Grant DMS-1151252 and FRG Grant DMS-1463901. Max Lieblich was partially supported by NSF CAREER Grant DMS-1056129, NSF Grant DMS-1600813, and a Simons Foundation Fellowship.
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Antieau, B., Auel, A., Ingalls, C. et al. Period-index bounds for arithmetic threefolds. Invent. math. 216, 301–335 (2019). https://doi.org/10.1007/s00222-019-00860-x
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DOI: https://doi.org/10.1007/s00222-019-00860-x