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On the size of the genus of a division algebra. (English) Zbl 1356.16013

Proc. Steklov Inst. Math. 292, 63-93 (2016) and Tr. Mat. Inst. Steklova 292, 69-99 (2016).
Let \(K\) be a field, \(\mathrm{Br}(K)\) its Brauer group, \(d(K)\) the class of finite-dimensional central division \(K\)-algebras, and for each \(n\in\mathbb N\), let \(_{n}\mathrm{Br}(K) = \{b_ p\in\mathrm{Br}(K): nb _ p = 0\}\) be the \(n\)-torsion part of \(\mathrm{Br}(K)\). Fix a separable closure \(K _{\mathrm{sep}}\) of \(K\), and for each \(D\in d(K)\), denote by \(M(D)\) the set of those extensions of \(K\) in \(K_{\mathrm{sep}}\), which are \(K\)-isomorphic to maximal subfields of \(D\); also, let \([D]\) be the equivalence class of \(D\) in \(\mathrm{Br}(K)\), \(\deg(D)\) the degree of \(D\), and \(\exp(D)\) the exponent of \(D\), i.e., the order of \([D]\) in \(\mathrm{Br}(K)\). By a genus of \(D\), the authors mean the set \(\mathbf {gen}(D)=\{[D']\in\mathrm{Br}(K):D'\in d(K)\), \(M(D')=M(D)\}\). The paper under review gives a detailed and effective proof of the assertion (stated as Theorem 1 by the authors, and announced in [the first author et al., C. R., Math., Acad. Sci. Paris 350, No. 17–18, 807–812 (2012; Zbl 1272.16020)] that \(\mathbf{gen}(D)\) is finite, provided that \(K\) is a finitely-generated field, \(D\in d(K)\) and \(\deg(D)\) is not divisible by the characteristic \(\mathrm{char}(K)\) (concerning the restriction on \(K\), see [the first author et al., Russ. Math. Surv. 70, No. 1, 83–112 (2015; Zbl 1325.12006); translation from Usp. Mat. Nauk. 70, No. 1, 89–122 (2015)]. When \(K\) is a global field, e.g., a finite extension of the field \(\mathbb Q\) of rational numbers, the conclusion of Theorem 1 holds, for every \(D \in d(K)\); this is an immediate consequence of Fein’s description of \(\mathbf{gen}(D)\) (see [B. Fein, Proc. Am. Math. Soc. 32, 427–429 (1972; Zbl 0219.16012)]), obtained from the part of class field theory concerning \(\mathrm{Br}(K)\).
The proof of Theorem 1 relies on Theorem 2 of this paper, stated as follows: For each \(n\in\mathbb N\), there exists a nonempty set \(V_n\) of discrete valuations of \(K\) satisfying the following conditions: (A) For any \(a \in K ^ {\ast }\), the set \(V _ n(a) = \{v \in V _ n: v(a) \neq 0\}\) is finite; (B) For any \(v \in V _ n\), \(\mathrm{char}(K ^ {(v)})\) does not divide \(n\), where \(K ^ {(v)}\) is the residue field of \((K, v)\); in particular, the residue map \(\rho _ v: \;_ n\mathrm{Br}(K) \to \mathrm{Hom}(\mathcal{G}^ {(v)}, \mathbb Z/n\mathbb Z)\) is defined (\(\mathcal{G}^ {(v)}\) denotes the absolute Galois group of \(K ^ {(v)}\)); (C) The unramified Brauer group \(_ n\mathrm{Br}(K)_ {V _{n}}\) with respect to \(V _ n\), defined as the intersection of the kernels ker\(\rho _ v\), \(v \in V _ n\), is finite. As shown in an earlier paper by the authors [Bull. Math. Sci. 3, No. 2, 211–240 (2013; Zbl 1293.16015)], Theorem 2 applied to the case of \(n = \deg(D)\), ensures the following: the set \(R(D)\) of those \(v \in V _ n\) where \(D\) ramifies is finite; \(\mathbf{gen}(D)\) is finite and its order \(|\mathbf{gen}(D)| \) is at most equal to \(\varphi (n) ^ r| _ n\mathrm{Br}(K)_ {V_{n}}| \), \(\varphi \) and \(r\) being the Euler function and the cardinality of \(R(D)\), respectively. The proof of Theorem 2 contains several results of independent interest. An alternative approach to the proof of Theorem 2, suggested by Colliot-Thélène and based on étale cohomology, is outlined in an appendix.

MSC:

16K20 Finite-dimensional division rings
16K50 Brauer groups (algebraic aspects)
11R52 Quaternion and other division algebras: arithmetic, zeta functions
12E15 Skew fields, division rings
12F20 Transcendental field extensions
14H05 Algebraic functions and function fields in algebraic geometry

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