Jean-Claude Douai

Algebraic covers : field of moduli versus field of definition

Abstract The field of moduli of a finite cover f: X → B a priori defined over the separable closure Ks of a field K, with B defined over K, need not be a field of definition. This paper provides a cohomological measure of the obstruction. The case of G-covers, i.e., Galois covers given together with their automorphisms, was fairly well-known. But no such cohomological measure was available for mere covers. In that situation, the problem is shown to be controlled not by one, as for G-covers, but by several characteristic classes in H2(Km, Z(G)), where Km is the field of moduli and Z(G) is the center of the group of the cover. Furthermore our approach reveals a more hidden obstruction coming on top of the main one, called the first obstruction and which does not exist for G-covers. In contrast with previous works, our approach is not based on Weils descent criterion but rather on some elementary techniques in Galois cohomology. Furthermore the base space B can be an algebraic variety of any dimension and the ground field K a field of any characteristic. Our main result yields concrete criteria for the field of moduli to be a field of definition. Our main result also leads to some local-global type results. For example we prove this local-to-global principle: a G-cover f: X → B is defined over ℚ if and only if it is defined over ℚp for all primes p.

Gerbes and covers

We use the theory of gerbes to provide a more conceptual approach to questions about models of a cover and their fields of definition.

Local-global principles for algebraic covers

This paper is devoted to some local-global type questions about fields of definition of algebraic covers. Letf:X→B be a covera priori defined over. Assume that the coverf can be defined over each completion ℚ{p} of ℚ. Does it follow that the cover can be defined over ℚ? This is thelocal-to-global principle. It was shown to hold for G-covers [DbDo], i.e., for Galois covers given with their automorphisms. Here we prove that, in the situation ofmere covers, the local-to-global principle holds under some additional assumptions on the groupG of the cover and the monodromy representationG→Sd (withd=deg(f)). This local-to-global problem is closely related to the obstruction to the field of moduli being a field of definition. This problem was studied in [DbDo], which is the main tool of the present paper.