Cohomological Approaches to Rational Points This Workshop Will Be

Cohomological approaches to rational points

This workshop will be organized by Fedor Bogomolov, Antoine Chambert- Loir, Jean-Louis Colliot-Thélène (chair), A. Johan de Jong, Raman Parimala and will be held March 27–31, 2006 at MSRI. • Etale´ cohomology of varieties over number fields Varieties over a number field may have rational points in each comple- tion of the field without having a rational point over the number field itself. As explained by Manin in 1970, the Brauer group accounts for many such examples. There are here two directions of research. In one direction, one tries to produce classes of varieties for which the Brauer-Manin condition is the only condition for the existence of a rational point. It is an open question whether rationally connected varieties consti- tute such a class. A case classically studied, but still of interest, is that of homogeneous spaces of algebraic groups. In recent times, much work has been devoted to one parameter families of homogeneous spaces of algebraic groups (linear or not). Research here ties in with research on the variation of the rank in a one-parameter family of abelian varieties, as well as on the variation of the Selmer group in a family. In another direction, Skorobogatov has shown that for some varieties the Brauer-Manin conditions may be satisfied but there is no rational point. The tool used up till now is descent by means of torsors under noncommutative, disconnected groups (as for torsors under commutative groups or connected groups, they essentially give the same information as the Brauer-Manin conditions). One does not believe that torsors produce the only obstruction to the existence of rational points. Indeed, it has been shown that such a statement would contradict well-known conjectures of Lang and others on rational points on varieties of general type. Even the case of curves is open. There are interesting analogues of the Brauer-Manin condition over other types of (semi-)global fields, involving higher dimensional cohomology. This ties in with higher class field theory àla Kato-Saito.

1 • Etale cohomology of varieties over finite fields The very powerful methods developed by Deligne to establish the Weil conjectures still find applications. Let us here mention Esnault’s proof that any (smooth, projective) rationally connected variety over a finite field has a rational point, and her recent proof that such is also the case for the (possibly singular) specialization of a (smooth, projective) rationally connected variety over a p-adic field. Such techniques also play a rôle in the study of the rank of elliptic curves in a family. • Coherent cohomology, obstruction methods The curve deformation techniques which play such an important rôle in Mori’s classification programme for higher dimensional algebraic varieties over the complex field have been applied to problems with an arithmetical flavour. Let us mention here the Graber-Harris-Starr theorem that rationally connected varieties over a function field in one variable over the complex fields always have a rational point, the “converse” theorem by the same authors, de Jong’s theorem that index coincides with exponent for central simple algebras over a function field in two variables over the complex field, Kollár’s theorem that over a p-adic field, there are only finitely many R- equivalence classes on the set of rational points of a rationally connected variety, finally the theorem by Kollár and Szabóthat if such a variety over a p-adic field is rationally connected and has good reduction, then the reduced Chow group of zero-cycles is trivial. An open question here is whether any smooth, projective rationally connected variety over a p-adic field acquires a rational point after an unramified extension of the p-adic field. This is an arithmetic analogue of the Graber-Harris-Starr theorem. Another open question is whether the reduced Chow group of zero-cycles on a variety over a p-adic field is always finite. One is also interested in systematic ways of detecting nontrivial elements in such a group. The case of surfaces is controlled by algebraic K-theory. The higher dimensional case is much of a

2 mystery. A vague but stimulating question is whether there is an analogue of the Graber-Harris-Starr theorem for suitable varieties over function fields of two variables over the complex field. Over function fields of one variable over the complex field, an interesting open question is whether rationally connected varieties satisfy weak approximation. Another question of interest here is the geometric characterization of varieties which satisfy “potential density”, i.e. over a finite extension of their field of definition rational points are dense for the Zariski topology. Ratio- nally connected varieties presumably satisfy this property, as well as a much broader class of varieties. • Algebraic geometry over Q; Arakelov geometry Diophantine approximation, within or without Arakelov geometry, has provided powerful tools and important theorems concerning the existence of rational points on algebraic varieties. Let us only mention the theorem of Roth, Vojta’s proof of Mordell’s conjecture, Faltings’s subsequent proof of Lang’s conjecture about rational points on subvarieties of abelian varieties. The ideas and the techniques involved have important but still mysterious connections with Nevanlinna theory and hyperbolicity. Similar techniques, as initiated by Chudnovsky, also allowed to elucidate some cases of the Grothendieck–Katz conjectures concerning algebraic solutions of dif- ferential equations (André) or algebraic leaves of foliations (Bost). The distribution of algebraic points on varieties is also a theme of interest. Arakelov geometry has provided tools which enabled Ullmo and Zhang to solve Bogomolov’s conjecture on the distribution of algebraic points of small height on subvarieties of abelian varieties. This approach has been generalized very recently to more general contexts, e.g. dynamical systems of an arithmetic origin, and also distribution for the p-adic topology, using Berkovich spaces. The André-Oort conjecture which is an analogue of Bo- gomolov’s conjecture on Shimura varieties has also seen recent spectacular developments (work of Clozel–Ullmo and Edixhoven–Yafaev).