Appears in collection : Model theory of valued fields / Théorie des modèles des corps valués

The model theory of valued fields usually concerns a single valuation, or a family of independent valuations. But many phenomena of geometry and number theory are ‘global' in nature and invisible to any finite number of places. Much is ultimately based on a single relation among the different valuations and absolute values, the product formula. This formula can be axiomatized within continuous logic as the theory GVF. I will describe some of what we know and some of what we don't know about this theory. In particular I hope to reach the role of certain type-definable subgroups of abelian varieties, intriguingly analogous to Manin kernels in differential algebra, the associated interpretable Hilbert spaces, the Hodge index theorem, and their roles in proving stability of certain fragments. This will be in part introductory, and in part complementary, to talks by Ben Yaacov and Szachniewicz.