Rational and integral points on higher-dimensional varieties The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop “Rational and integral points on higher-dimensional varieties.” This material is not for public distribution. Corrections and new material are welcomed and can be sent to [email protected] Version: Fri May 9 09:47:46 2014 1 2 Table of Contents A.ParticipantContributions . 3 1. Bright, Martin 2. Browning, Tim 3. Colliot-Thelene, Jean-Louis 4. Derenthal, Ulrich 5. Frei, Christopher 6. Harari, David 7. Hassett, Brendan 8. Heath-Brown, Roger 9. Loughran, Daniel 10. Lucchini Arteche, Giancarlo 11. Peyre, Emmanuel 12. Salberger, Per 13. Sofos, Efthymios 14. Testa, Damiano 15. Tschinkel, Yuri 16. Wittenberg, Olivier 17. Wooley, Trevor 18. Yasufuku, Yu 3 Chapter A: Participant Contributions A.1 Bright, Martin I am interested in the relationship between the Brauer–Manin obstruction and the geometry of models of varieties. I would like to suggest the following problems for consider- ation. It is now well known that the absence of a Brauer–Manin obstruction to the Hasse principle on a variety can often be explained by a local condition at one place. Such a condition often naturally comes from the geometry of a model at that place. For example, let X be a cubic surface over a number field k. If there is a place of k where X has a regular model whose special fibre is a cone, then there is no Brauer–Manin obstruction to the Hasse principle on X. According to the widely-believed conjecture of Colliot-Th´el`ene and Sansuc, this should imply that X satisfies the Hasse principle. How might one try to prove this? Can any existing techniques make use of the fact that X reduces to a cone at one place? A related question is the following. Given a cubic surface X over a number field, taking a quadratic extension of the base field affects neither whether X has a rational point nor whether there is a Brauer–Manin obstruction to the Hasse principle on X. What kind of models can we find for a cubic surface over a DVR if we are allowed to take quadratic base extensions? A.2 Browning, Tim I am interested in questions about integral and rational points on varieties. My back- ground is analytic number theory, which means that I am particularly drawn to quantitative problems, or to qualitative problems that can be tackled using analytic methods. One area that currently interests me is the following question: how frequently do failures of the Hasse principle or weak approximation occur in a given family of varieties over Q? In recent work with R´egis de la Bret`eche (arXiv:1210.4010) I looked at the family of Chˆatelet surfaces Xa,b,c,d given by the affine equation y2 + z2 =(at2 + b)(ct2 + d). Here the Brauer group Br(Xa,b,c,d)/Br(Q) is non-trivial and we obtain precise asymptotics, as H → ∞, for the number of surfaces Xa,b,c,d of height H which fail the Hasse principle. The main result shows, in particular, that 100% of the surfaces satisfy the Hasse principle. Is there an interpretation of the exponents of H and log H that we obtain in our asymptotic formula? For this it would be useful to have more examples where such asymptotics can be proven. More generally, given an arbitrary family X → Y defined over Q, under what conditions can we expect 100% of the rational points y ∈ Y (Q) to produce fibres Xy which satisfy the Hasse principle? I am also interested in the Hardy–Littlewood circle method in various forms. For example, in work in preparation with Pankaj Vishe, I have started looking at the circle method over Fq(t). We plan to use a “Kloosterman refinement” to study smooth cubic hypersurfaces in sufficiently many variables. This ought to lead to weak approximation results and information about the irreducibility of the moduli space of rational curves on cubic hypersurfaces defined over Fq. I would be interested in learning more about how to set up the circle method over Qp(t), and moreover, thinking about how one might profitably do the same for Q(t). 4 In recent work with Matthiesen (arXiv:1307.7641) I have been involved in proving that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for smooth compactifications of affine equations r mi NK/Q(x1,...,xn)= c Y(t − ei) , i=1 where K/Q is any number field of degree n, m1,...,mr ∈ Z≥1 and e1,...,er ∈ Q are pairwise distinct. The novelty of this is that it combines the descent theory of Colliot-Th´el`ene and Sansuc with new input from additive combinatorics to study the system of equations (i) (i) NK/Q(x1 ,...,xn )= ci(u − eiv), (1 ≤ i ≤ r), in Arn+2. What are the most general statements that this influx of new technology is capable of producing? A.3 Colliot-Thelene, Jean-Louis 1) Lien entre obstruction de Brauer-Manin enti`ere et obstruction de Brauer-Manin rationnelle. 4 Soient a,b,c ∈ Z avec c.(a − b) =6 0. Consid´erons dans AZ avec coordonn´ees (x,y,z,t) le ferm´e V d´efini par y2 = c(z − at)(z − bt), zt = x2 et l’ouvert U ⊂ V compl´ementaire de (0, 0, 0, 0)Z. La Q-vari´et´e U = UQ est le cˆone ´epoint´e 3 sur la courbe C de genre 1 dans PQ d´efinie par les mˆemes ´equations. U Z Br(U) ∅ Q Br(C) 6 ∅ Peut-on avoir [Qp∪∞ ( p)] = mais [Qp∪∞ C( p)] = ? [L’alg`ebre A d´efinie par recollement des alg`ebres de quaternions (z − at,t)=(z − at,z)=(c(z − bt),t)=(c(z − bt),z) est dans Br(U) mais ne provient pas de Br(C).] 1 2) Soit f : X → AQ un Q-morphisme. Y a-t-il des restrictions au type de comporte- ment quantitatif asymptotique du nombre de points de f(X(Q)) ? A.4 Derenthal, Ulrich I am particularly interested in the use of universal torsors and Cox rings to study rational and integral points on higher-dimensional varieties. Using the universal torsor method, I have studied Manin’s conjecture for del Pezzo surfaces, first over Q and then over imaginary-quadratic fields. Now I am particularly inter- ested in the extension to arbitrary number fields (cf. recent work of Frei and Pieropan) and in higher-dimensional cases. Furthermore, I have used universal torsors and the fibration method to study the Hasse principle for rational and integral points and weak and strong approximation, in particular on “normic bundles”. Here I am currently particularly interested in the methods from additive combinatorics recently introduced by Browning, Matthiesen, and Skorobogatov. 5 A.5 Frei, Christopher At the moment, I am very interested in the distribution of rational points on (almost) Fano varieties, mostly del Pezzo surfaces. My perspective on these questions comes from Minkowski’s geometry of numbers, in particular lattice point counting. One prevalent method to prove Manin’s conjecture for varieties X of small dimension (which are out of reach of the Hardy-Littlewood circle method) is to parameterize the rational points on the variety by integral points on a universal torsor and to count these integral points by lattice point counting methods from analytic number theory and the geometry of numbers. This approach has been successfully applied to many special cases, mostly del Pezzo surfaces, over the rational numbers Q. The passage from Q to arbitrary number fields K is not straightforward. One of the reasons for this is that the parameterization obtained from the universal torsor is modulo an action of the group of OK -points of the Neron-Severi torus of X, where OK is the ring of × r integers of K. In the split case this is an action of (OK ) , where r is the Picard number of × X. Over Q, this action is easy to deal with, but as soon as OK is infinite, one needs to work with a suitable fundamental domain. Such a fundamental domains can be constructed quite naturally, motivated by ideas of S. H. Schanuel, but it seems hard to combine this construction with many of the analytic tools applied to the corresponding lattice point counting problems over Q. In recent joint work with M. Pieropan, we proved Manin’s conjecture for a specific del Pezzo surface of degree 4 and type A3 + A1 over an arbitrary number field K. We extended the analytic approach applied by U. Derenthal over Q by a higher-dimensional counting argument based on a recent result of F. Barroero and M. Widmer on counting lattice points in families definable in an o-minimal structure. In principle, this approach seems to be applicable to many other cases, with the challenge being to keep the error terms sufficiently small. One way to achieve this would be to make Barroero and Widmer’s result compatible with the usual analytic approaches to achieve error-cancellation, but I do not know (yet) how feasible this would be. During the workshop, I hope to have the opportunity to discuss these problems with experts. Moreover, I hope to broaden my perspective on the distribution of rational points by learning about the other methods that have been successfully applied to Manin’s conjecture, and about interesting (classes of?) higher-dimensional (almost) Fano varieties to study. I would also be interested in potential applications of lattice point counting methods to other related problems, e.g.