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On elementary equivalence, isomorphism and isogeny

Pete L. Clark — 2006

Journal de Théorie des Nombres de Bordeaux

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...

The period-index problem in WC-groups IV: a local transition theorem

Pete L. Clark — 2010

Journal de Théorie des Nombres de Bordeaux

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 of torsors under abelian varieties with good reduction and a generalization...

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