Towards a model theory of global fields

Résumé du cours

Beyond the sum of the local geometries associated with each place of a field, there is a fascinating global geometry arising from their interaction. As recognized by Artin‑Whaples in the discrete setting, this interaction is governed by a simple relation among the places, the product formula. I will describe a model‑theoretic framework, developed jointly with Itai Ben Yaacov, able to axiomatize this relation. We will see that an even preliminary model‑theoretic investigation leads to deep algebro‑geometric structures associated with cones of divisors and curves. This class will concentrate on the purely non‑archimedean case; the field $\Cc(t)^{alg}$, with a height function into $\Rr$, is an example. We will show that it is existentially closed. The first half of the class will be an introduction to the required elements of convexity theory and algebraic geometry.