Pierre Dèbes

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.

Indecomposable polynomials and their spectrum

k is said to be indecomposable in k[x] if it is not of the form u(H(x)) with H(x)2 k[x] and u2 k[t] with deg(u) 2. An element 2 k is called a spectral value of F (x) if F (x) is reducible in k[x]. It is well-known that (1) F (x)2 k[x] is indecomposable if and only if F (x) is irreducible in k( )[x] (where is an indeterminate), (2) if F (x)2 k[x] is indecomposable, then the subset sp(F ) k of all spectral values of F (x)|the spectrum of F (x)|is nite ; and in the opposite case, sp(F ) = k, (3) more precisely, if F (x) 2 k[x] is indecomposable and for every 2 k, n( ) is the number of irreducible factors of F (x) in k[x], then (F ) := P 2k (n( ) 1) deg(F ) 1. In particular, card(sp(F )) deg(F ) 1. Statement (3), which is known as Stein’s inequality, is due to Stein [13] in characteristic 0 and Lorenzini [10] in arbitrary characteristic (but for two variables); see [11] for the general case. This paper oers some new results in this context. Inx2, given an indecomposable polynomial F (x) with coecients

Specializations of Galois covers of the line

The main topic of the paper is the Hilbert‐Grunwald property of Galois covers. It is a property that combines Hilbert’s irreducibility theorem, the Grunwald problem and inverse Galois theory. We first present the main results of our preceding paper which concerned covers over number fields. Then we show how our method can be used to unify earlier works on specializations of covers over various fields like number fields, PAC fields or finite fields. Finally we consider the case of rational function fields κ(x) and prove a full analog of the main theorem of our preceding paper.

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.

Irreducibility of Hypersurfaces

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P)2 − 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.

On the Malle conjecture and the self-twisted cover

For a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant |dE| ≤ y is shown to grow at least like a power of y, for some specified positive exponent. The groups G are the regular Galois groups over Q and the counted extensions E/Q are obtained by specializing a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime p ≤ log(y)/δ for some δ ≥ 1. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bounded discriminant. The local conditions further make it a notable constraint on regular Galois groups over Q. The method uses a new version of Hilbert’s irreducibility theorem that counts the specialized extensions and not just the specialization points. A main tool for it is the self-twisted cover that we introduce.

HARBATER–MUMFORD COMPONENTS AND TOWERS OF MODULI SPACES

A method of choice for realizing finite groups as regular Galois groups over Q(T ) is to find Q-rational points on Hurwitz moduli spaces of covers. In another direction, the use of the so-called patching techniques has led to the realization of all finite groups over Qp(T ). Our main result shows that, under some conditions, these p-adic realizations lie on some special irreducible components of Hurwitz spaces (the so-called Harbater-Mumford components), thus connecting the two main branches of the area. As an application, we construct, for every projective system (Gn)n≥0 of finite groups, a tower of corresponding Hurwitz spaces (HGn )n≥0, geometrically irreducible and defined over Q, which admits projective systems of Qur p -rational points for all primes p not dividing the orders |Gn| (n≥0). 2000 MSC. Primary 12F12 14H30 14H10 ; Secondary 14D15 14G22 32Gxx

Hilbert subsets ands-integral points

A classical tool for studying Hilberts irreducibility theorem is Siegels finiteness theorem forS-integral points on algebraic curves. We present a different approach based ons-integral points rather thanS-integral points. Given an integers>0, an elementt of a fieldK is said to bes-integral if the set of placesv ∈MK for which |t|v > l is of cardinality ≤s (instead of contained inS for “S-integral”). We prove a general diophantine result fors-integral points (Th.1.4). This result, unlike Siegels theorem, is effective and is valid more generally for fields with the product formula. The main application to Hilberts irreducibility theorem is a general criterion for a given Hilbert subset to contain values of given rational functions (Th.2.1). This criterion gives rise to very concrete applications: several examples are given (§2.5). Taking advantage of the effectiveness of our method, we can also produce elements of a given Hilbert subset of a number field with explicitely bounded height (Cor.3.7). Other applications, including the case thatK is of characteristicp>0, will be given in forthcoming papers ([8], [9]).

Resultats Recents Lies Au Theoreme D’Irreductibilite de Hilbert

Le theoreme d’irreuctibilite de Hilbert est un reltat de la fin du siele dernier [17]. Le probleme est le suivant: etant donnes un corps k et Pl ,…, Pn n polynomes irreductibles dans k(X1,..., Xr)[Y1 ,..., Ys] , montrer que l’ensemble qu’on note classiquement Hk(P1 ,..., Pn), constitue des specialisations (xl , x2 ,..., xr) des indeterminees (X1 ,..., Xr) pour lesquelles les polynomes Pi(xl ,..., xr , Y1 ,..., Ys), i= 1,2 ,..., n, sont irreductibles dans k[Y1 ,..., Ys], contient beaucoup d’elements de kr. Precisement, on appelle partie hilbertienne de kr tout ensemble, intersection d’un ensemble du type Hk(P1 ,..., Pn) avec un ouvert de Zariski de kr et on dit que le corps k est hilbertien si pour tout entier r≥1, les parties hilbertiennes sont non vides. On appelle aussi ensemble mince tout ensemble dont le complementaire contient une partie hilbertienne. En ces termes, le theoreme d’irreductibilite de Hilbert s’enonce: