Short Courses

Fields of Definition of Covers; Embedding Problems over Large Fields. Pierre Dèbes.

I. The first talk deals with joint work with Jean–Claude Douai. Let f : X → B be a finite cover defined over the separable closure Ks of a field K, withB an algebraic variety defined over K. Assume that f is isomorphic to each of its conjugates under G(Ks/K). The field K is called the field of moduli of the cover. Does it follow that the given cover can be defined over K? The answer is “No” in general: there is an obstruction to the field of moduli being a field of definition. Still, how can the obstruction be measured? We present a general approach for this problem. The obstruction is entirely of a cohomological nature. This was known only in the case of G–covers,i.e., Galois covers given together with their automorphisms. This special case happens to be the simplest one. In the situation of mere covers, the problem is shown to be controlled not by one, as for G–covers, but by several characteristic classes in H2(K, Z(G), L) (for a certain action L of G(Ks/K) on the center Z(G) 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.

Our Main Theorem yields quite concrete criteria for the field of moduli to be a field of definition. Such criteria were not available in the general situation of mere covers.