Let k be an algebraically closed field of arbitrary characteristic, let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime , the absolute Galois group of K acts on the -adic etale cohomology modules of X. We prove that this family of representations varying over is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations. 0 2010 MSC: 11G10, 14F20. 0

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A variety X over a field K is of Hilbert type if X(K) is not thin. We prove that if f : X → S is a dominant morphism of K-varieties and both S and all fibers f −1 (s), s ∈ S(K), are of Hilbert type, then so is X. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

Communicated by E.M. Friedlander MSC: 11E30; 11G10; 14K15 a b s t r a c tAn abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on ℓ-torsion points, for almost all primes ℓ, contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with the endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K . This generalizes a recent result of Chris Hall.

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.