From the viewpoint of geometric classification, there are two extreme cases of smooth varieties X defined over an algebraically closed field: those which are hyperbolic, and those which are rationally connected. If k is no longer algebraically closed, a central question of Algebraic Arithmetic Geometry is what properties of k force X to have a rational point in those two opposed cases. It is conjectured (Lang-Manin, extended by Kollár), that rationally connected varieties have a rational point over a C1 field. It has been shown for function fields by Graber-Harris-Starr and by myself over a finite field. There is no relation between their geometric proof relying on the geometry of the moduli of punctured curves and my proof relying on motivic analogies between Hodge level and slopes in l-adic cohomology. The study of the case of the maximal unramified extension of the p-adic numbers might provide a bridge through the use of the inertia. Very little is known on Grothendieck's section conjecture, which predicts that sections of the Galois group of k, assumed to be a finite type over Q, into the arithmetic fundamental group of X, are given by rational points. Our hope goes in two directions, arithmetic and geometric on one side, motivic on the other. With Wittenberg, we hope to use Beilinson's geometric description of the nilpotent completion of the fundamental group, and with Levine, we wish to characterize sections of the motivic groups of mixed Tate motives over k and X and relate this to the section conjecture.
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FinanzierungsplanERC-AG - ERC Advanced Grant