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On fields of moduli of curves

✍ Scribed by Pierre Dèbes; Michel Emsalem

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
795 KB
Volume
211
Category
Article
ISSN
0021-8693

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✦ Synopsis

The field of moduli K of a curve X a priori defined over the separable closure K s of K need not be a field of definition. This paper shows that the obstruction is essentially the same as the obstruction to K being a field of definition of the cover X ~ X/Aut(X). Using previous results of Dbbes-Douai, we then obtain a cohomological measure of the obstruction. This yields concrete criteria for the field of moduli to be a field of definition. An interesting application is the following local-global principle. If a curve X, together with all of its automorphisms, is defined over Qp for all primes p, then it is defined over Q.

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