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Period-index bounds for arithmetic threefolds. (English) Zbl 1436.14040

The main result of this article is Theorem 1.1, which gives the first uniform period-index bound for fields of transcendence degree 2 over local fields.
The period-index problem is classical in the theory of Brauer groups. For a survey of this problem, see section 4 of [A. Auel et al., Transform. Groups 16, No. 1, 219–264 (2011; Zbl 1230.16016)]. Given a Brauer class \(\alpha\) over a field, Wedderburn’s theorem implies that \(\alpha = [D]\) for a unique division algebra \(D\). The period of \(\alpha\), \(\mathrm{per}(\alpha)\), is its order in the Brauer group and its index, \(\mathrm{ind}(\alpha)\), is the degree of \(D\). It is well-known that \(\mathrm{per}(\alpha)\) divides \(\mathrm{ind}(\alpha)\) and that they have the same prime factors. The period-index problem is concerned with the smallest \(n\) so that \(\mathrm{ind}(\alpha)\) divides \(\mathrm{per}(\alpha)^n\).
Such a period-index bound was established in the following cases:
If \(k\) is a local or a global field, then \(n=1\) [R. Brauer et al., J. Reine Angew. Math. 167, 399–404 (1932; Zbl 0003.24404)].
If \(k\) is a finitely generated field of transcendence degree \(2\) over an algebraically closed field, then \(n=1\) [A. J. de Jong, Duke Math. J. 123, No. 1, 71–94 (2004; Zbl 1060.14025)].
If \(k\) is a field of transcendence degree \(1\) over a local field, then \(n=2\) [D. J. Saltman, J. Ramanujan Math. Soc. 12, No. 1, 25–47 (1997; Zbl 0902.16021)] and [R. Parimala and V. Suresh, Invent. Math. 197, No. 1, 215–235 (2014; Zbl 1356.11018)]. For analogous results over fields of transcendence degree \(1\) over higher local fields, see [M. Lieblich, J. Reine Angew. Math. 659, 1–41 (2011; Zbl 1230.14021)] and [D. Harbater et al., Invent. Math. 178, No. 2, 231–263 (2009; Zbl 1259.12003)].
If \(k\) is a field of transcendence degree \(2\) over a finite field, then \(n=2\) [D. Harbater et al., Invent. Math. 178, No. 2, 231–263 (2009; Zbl 1259.12003)] and [M. Lieblich, J. Reine Angew. Math. 659, 1–41 (2011; Zbl 1230.14021)].

These results support the following period-index conjecture (Conjecture 1.3 in the paper under review): Let \(k\) be an algebraically closed, \(C_1\), or \(p\)-adic field. Set \(e=0,1,2\) respectively. Let \(K\) be a field of transcendence degree \(n\) over \(k\). Then \[\mathrm{ind}(\alpha) | \mathrm{per}(\alpha)^{n-1+e}.\]
In the case of the function field of a \(p\)-adic surface \(S\), this conjecture predicts, that \(\mathrm{ind}(\alpha) | \mathrm{per}(\alpha)^3\). In the present paper, the authors show that \(\mathrm{ind}(\alpha) | \mathrm{per}(\alpha)^4\) for \(\alpha\) of period prime to \(6p\). The proof uses Gabber’s theory of prime-to-\(l\) alterations and the deformation theory of twisted sheaves.

MSC:

14F22 Brauer groups of schemes
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
16K50 Brauer groups (algebraic aspects)
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