We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin’s conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of motivic Euler product. It relies on a construction of Denef and Loeser which associates a virtual motive to a first order logic ring formula.

Denef and Loeser denned a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.

In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

In this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

For a linear group $G$ acting on an absolutely irreducible variety $X$ over $\mathbb{Q}$, we describe the orbits of $X(\mathbb{Q}_p)$ under $G(\mathbb{Q}_p)$ and of $X(\mathbb{F}_p((t)))$ under $G(\mathbb{F}_p((t)))$ for $p$ big enough. This allows us to show that the degree of a wide class of orbital integrals over $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$ is less than or equal to $0$ for $p$ big enough, and similarly for all finite field extensions of $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$.