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The period-index problem in WC-groups. IV: A local transition theorem. (English. French summary) Zbl 1258.11094
This clearly written and instructive paper gives an excellent survey of the local period-index problem from the geometric perspective, i.e. that of [D. Lorenzini, Q. Liu and M. Raynaud, Invent. Math. 157, No. 3, 455–518 (2004; Zbl 1060.14037)] and [O. Gabber, Q. Liu and D. Lorenzini, “The index of an algebraic variety”, Invent. Math. 192, No. 3, 567–626 (2013) doi:10.1007/s00222-012-0418-z] (cf. §4 of the paper under review). The main contribution appears to be introduction of a certain period-index obstruction map in flat cohomology (§2), which is used to prove the following result. Let \(K\) be a field which is complete with respect to a discrete valuation. Let \(k\) denote the residue field of \(K\), which is assumed to be perfect. Let \(K^{\text{sep}}\) denote a separable closure of \(K\), and \(\mathfrak{g}_K\) the associated Galois group \(\text{Gal}(K^{\text{sep}}/K)\). In general, given \(M\) a \(\mathfrak{g}_K\)-module, and \(\eta\) a class in the Galois cohomology group \(H^1(K, M) = H^1(\mathfrak{g}_K, M)\), write \(P(\eta)\) to denote the period of \(\eta\), and \(I(\eta)\) to denote the index of \(\eta\) (as defined e.g. in [P. Clark, “The period-index problem for WC-groups. II: Abelian varieties”, preprint http://arxiv.org/abs/math/0406135]).
(a) Suppose that for some integer \(i \geq 1\) and some function \(c: {\mathbb{Z}}_{\geq 0} \longrightarrow {\mathbb{Z}}_{\geq 0},\) the following property holds: For all abelian varieties \(A/k\) and for all classes \(\eta \in H^1(k, A)\), \(I(\eta) \leq c(\dim A) P(\eta)^i\). Then, there exists a function \(C: {\mathbb{Z}}_{\geq 0} \longrightarrow {\mathbb{Z}}_{\geq 0}\) such that the following property holds: For all finite extensions \(L/K\), for all principally polarized abelian varieties \(A/L\), and for all classes \(\eta \in H^1(L, A)\), \(I(\eta) \leq C(\dim A) P(\eta)^{\dim A + i}\).
(b) Assume that \(k\) has characteristic 0. Suppose that for some integer \(i \geq 1\) and some function \(c: {\mathbb{Z}}_{\geq 0} \longrightarrow {\mathbb{Z}}_{\geq 0},\) the following property holds: For all finite extensions \(l/k\), for all nontrivial abelian varieties \(A/l\), and for all classes \(\eta \in H^1(l, A)\), \(I(\eta) \leq c(\dim A) P(\eta)^{\dim A + i -1}\). Then, there exists a function \(C: {\mathbb{Z}}_{\geq 0} \longrightarrow {\mathbb{Z}}_{\geq 0}\) such that the following property holds: For all finite extensions \(L/K\), for all principally polarized abelian varieties \(A/L\), and for all classes \(\eta \in H^1(L, A)\), \(I(\eta) \leq C(\dim A) P(\eta)^{\dim A + i}\).
For Part I of this series of papers see J. Number Theory 114, No. 1, 193–208 (2005; Zbl 1087.11036).

11R34 Galois cohomology
11G05 Elliptic curves over global fields
12G05 Galois cohomology
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