Research

Articles/Preprints

Higher amalgamation in \(\mathrm{ACFA}^{+}\)

Abstract:

We show two results on higher amalgamation in the theory \(\mathrm{ACFA}^{+}\), the model companion of the theory of difference fields with an additive character (added as a continuous logic predicate) on the fixed field in characteristic 0. On one hand, we show that the non-trivial condition for 3-amalgamation established in a preceding paper is not sufficient for 4-amalgamation. On the other hand, we show that when working over substructures whose \(\mathcal{L}_{\sigma}\)-reduct is a model of \(\mathrm{ACFA}\), \(n\)-amalgamation holds for all \(n\geq 3\).

Model theory of difference fields with an additive character on the fixed field

Abstract:

Following a research line proposed by Hrushovski in his work on pseudofinite fields with an additive character, we investigate the theory \(\mathrm{ACFA}^{+}\) which is the model companion of the theory of difference fields with an additive character on the fixed field added as a continuous logic predicate. \(\mathrm{ACFA}^{+}\) is the common theory (in characteristic 0) of the algebraic closure of finite fields with the Frobenius automorphism and the standard character on the fixed field and turns out to be a simple theory. We fully characterise 3-amalgamation and deduce that the connected component of the Kim-Pillay group (for any completion of \(\mathrm{ACFA}^{+}\)) is abelian as conjectured by Hrushovski. Finally, we describe a natural expansion of \(\mathrm{ACFA}^{+}\) in which geometric elimination of continuous logic imaginaries holds.

Pseudofinite fields with additive and multiplicative character

Abstract:

We introduce the theory \(\mathrm{PF}^{+,\times}\) of pseudofinite fields with generic additive and multiplicative character added as continuous logic predicates. Using the Weil bounds on character sums over finite fields as well as the Erdős-Turàn-Koksma inequality we show that it is the asymptotic theory (in characteristic 0) of finite fields with (sufficiently generic) additive and multiplicative character. Moreover, we establish quantifier elimination in a natural definitional expansion of the language and deduce that integration by the Chatzidakis-van den Dries-Macintyre counting measure is uniformly definable in the parameters. Finally, we show that \(\mathrm{PF}^{+,\times}\) is a simple theory.

An Approximate AKE Principle for Metric Valued Fields (with Martin Hils)

Abstract:

We study metric valued fields in continuous logic, following Ben Yaacov’s approach, thus working in the metric space given by the projective line. As our main result, we obtain an approximate Ax-Kochen-Ershov principle in this framework, completely describing elementary equivalence in equicharacteristic 0 in terms of the residue field and value group. Moreover, we show that, in any characteristic, the theory of metric valued difference fields does not admit a model-companion. This answers a question of Ben Yaacov.

PhD Thesis

Title:

The model theory of difference fields with an additive character on the fixed field