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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).

##### MSC:
 11R34 Galois cohomology 11G05 Elliptic curves over global fields 12G05 Galois cohomology
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