Vol. 5, No. 4, 2011

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Global descent obstructions for varieties

Jean-Marc Couveignes and Emmanuel Hallouin

Vol. 5 (2011), No. 4, 431–463

We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having Q as field of moduli, that admit models over every completion of Q, but have no model over Q.

algebraic curve, global field, descent, field of definition, field of moduli, stack
Mathematical Subject Classification 2000
Primary: 11R34
Secondary: 12G05, 11G35, 14D22
Received: 11 September 2009
Revised: 22 November 2010
Accepted: 19 February 2011
Published: 21 December 2011
Jean-Marc Couveignes
Institut de Mathématiques de Bordeaux
Université Bordeaux I et CNRS
161, cours de la Libération
F-33405 Talence
Emmanuel Hallouin
Département de mathématiques et informatique
Université de Toulouse le Mirail
5, allées Antonio Machado
F-31058 Toulouse