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Preprint Number 104

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104. Zoé Chatzidakis, Ehud Hrushovski
Difference fields and descent in algebraic dynamics - I
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Submission date: 24 November 2007

Abstract:

We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line. The paper comes in three parts. This first, semi-expository part contains some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens these results to arbitrary base fields and rational maps (where in part I we allow finite base change and correspondences.) Part III will include precise structure theorems for primitive algebraic dynamics that are not modular.

Mathematics Subject Classification: 03C60, 12H10, 14GXX

Keywords and phrases: model theory, difference fields, algebraic dynamics

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