Théorie des modèles des corps aux différences avec un caractère additif sur le corps fixe

Building on work of Hrushovski on pseudofinite fields with an additive character we study the model theory of difference fields (in characteristic 0) with an additive character on the fixed field added as a complex-valued continuous logic predicate. Their theory has a model companion, the theory ACFA+. It has quantifier elimination up to algebraically bounded quantifiers and coincides with the characteristic 0 asymptotic theory of the algebraic closure of finite fields together with the Frobenius and an additive character on the fixed field. The theory ACFA+ is simple but 3-amalgamation does not hold in general over aclσ-closed sets. Following a line of research suggested by Hrushovski, we give a complete characterisation of this phenomenon. It allows us to determine the identity component of the Kim-Pillay group as a topological group. In particular this confirms Hrushovski’s expectation that the group is abelian. In addition we deduce a classification of the (continuous logic-)imaginaries in ACFA+. We then study higher amalgamation. In contrast to the theories ACFA and PF+, we can construct a counterexample to 4-amalgamation over a set for which 3-amalgamation holds. Nevertheless, we show that n-amalgamation holds over models for all n ∈ N. In the last chapter of the thesis we generalise Hrushovski’s results in a different direction. Motivated by natural examples from number theory we introduce the theory PF+,× of pseudofinite fields with both, additive and multiplicative character (of infinite order). We show quantifier elimination in a natural extension of the language and obtain that PF+,× is the characteristic 0 asymptotic theory of finite fields with nontrivial additive and sufficiently generic multiplicative character. Then we prove that integration of definable predicates with respect to the Chatzidakis-Macintyre-van den Dries counting measure is uniformly definable in the parameters.