Fix an abstract field K. For each K-variety V, we will define an étale-open topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties.The étale-open topology is mostly of interest when K is large (also known as ample). On non-large fields, the étale-open topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some well-known facts about large fields, and classify the model-theoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with Chieu-Minh Tran, Erik Walsberg, and Jinhe Ye.
- Séminaire Géométrie et théorie des modèles