The period-index problem in WC-groups IV: a local transition theorem
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 583-606.

Soit K un corps de valuation discrète complet avec corps résiduel parfait k. En supposant des bornes supérieures pour la relation entre l’indice et la période pour des groupes de Weil-Châtelet sur k, nous déduisons des bornes supérieures correspondantes pour la relation entre l’indice et la période pour des groupes de Weil-Châtelet sur K. À une constante dépendant seulement de la dimension d’un torseur près, nous retrouvons des théorèmes de Lichtenbaum et Milne dans un contexte “sans dualité”. Nos techniques utilisent les modèles LLR des torseurs sous des variétés abeliennes avec bonne réduction et une généralisation de l’obstruction période-indice à la cohomologie plate. Dans un appendice, nous considérons des sujets apparentés relevant de l’arithmétique du corps.

Let K be a complete discretely valued field with perfect residue field k. Assuming upper bounds on the relation between period and index for WC-groups over k, we deduce corresponding upper bounds on the relation between period and index for WC-groups over K. Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a “duality free” context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and a generalization of the period-index obstruction map to flat cohomology. In an appendix, we consider some related issues of a field-arithmetic nature.

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DOI : https://doi.org/10.5802/jtnb.734
@article{JTNB_2010__22_3_583_0,
     author = {Pete L. Clark},
     title = {The period-index problem in {WC-groups} {IV:} a local transition theorem},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {583--606},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.734},
     zbl = {1258.11094},
     mrnumber = {2769333},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.734/}
}
Pete L. Clark. The period-index problem in WC-groups IV: a local transition theorem. Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 583-606. doi : 10.5802/jtnb.734. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.734/

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