Rational and Integral Points on Higher-Dimensional Varieties the American Institute of Mathematics

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

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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-Thélène 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 Régis de la Bretèche (arXiv:1210.4010) I looked at the family of Châtelet 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-Thélène 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 entière et obstruction de Brauer-Manin rationnelle. 4 Soient a,b,c ∈ Z avec c.(a − b) =6 0. Considérons dans AZ avec coordonnées (x,y,z,t) le fermé V défini par y2 = c(z − at)(z − bt), zt = x2 et l’ouvert U ⊂ V complémentaire de (0, 0, 0, 0)Z. La Q-variété U = UQ est le cône épointé 3 sur la courbe C de genre 1 dans PQ définie par les mêmes équations. U Z Br(U) ∅ Q Br(C) 6 ∅ Peut-on avoir [Qp∪∞ ( p)] = mais [Qp∪∞ C( p)] = ? [L’algèbre A définie par recollement des algèbres 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 interested 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. concerning integral points. A.6 Harari, David Cohomological obstructions to Hasse principle and weak approximation, in particular for homogeneous spaces of linear groups with arbitrary stabilizers. Density of rational points in real points in the case of finite (non-abelian) stabilizers.

Extension to fields of higher cohomological dimension (e.g. Qp(t), function fields of curves over quasi-finite fields). 6 A.7 Hassett, Brendan I am interested in the structure of rational curves on K3 surfaces, the behavior of rational points under derived equivalences of K3 surfaces, and geometric interpretations of moduli spaces of K3 surfaces with level structure. A.8 Heath-Brown, Roger My main interests in this area are: (1) Applications of the circle method to general hypersurfaces and other varieties. (2) The determinant method and its applications. (3) Manin’s conjecture. Ideally, I would like to understand cubic surfaces better! A.9 Loughran, Daniel My research interests revolve around the distribution and existence of rational points on varieties. In particular, my work so far has dealt with Manin’s conjecture on rational points of bounded height and the Brauer-Manin obstruction to the Hasse principle and weak approximation. Recently, I have become interested in the quantitative arithmetic of families of varieties. For example given a family of varieties over a number field, how many varieties in the family contain a rational point? The simplest case is for families of conics parametrised by some variety X, say, equipped with a suitable height function, i.e. conic bundles over X. One may then attempt to count the number of rational points x of bounded height on X for which the conic over x contains a rational point. To any such conic bundle one may associate a quaternion algebra over the function field of X, and this quaternion algebra specialises to zero at a rational point if and only if the corresponding conic in the family contains a rational point. Serre [Ser90] proved upper bounds for the corresponding counting problem in the case where X = Pn using the large sieve, in particular he showed that 0% of the conics in the family contain a rational point as soon as the associated quaternion algebra is ramified. He also asked whether these bounds were sharp. I answered affirmatively Serre’s question in some special cases in my recent work [Lou13]. I would be interested in seeing more progress on Serre’s problem, i.e. obtaining lower bounds for the number of conics of bounded height which contain a rational point for suitable families. For example, the answer to Serre’s question seems to be unknown for Châtelet surfaces (viewed as conic bundles over the projective line). I am also interested in considering generalisations of Serre’s work to other classes of varieties, in particular finding other examples of families of varieties where one can show that 0% of the varieties in the family contain a rational point, even though there are still varieties in the family which contain a rational point. I am in the process of formulating some conjectures on such problems, which I began in [Lou13], in particular trying to find geometric conditions on the family which guarantee that 0% of the varieties in the family contain a rational point. I believe that this should be intimately related to the types of singular fibres which occur in the family. I would be interested in investigating such problems further at the workshop. Bibliography 7

[Lou13] D. Loughran, The number of varieties in a family which contain a rational point. Submitted, arXiv:1310.6219.

[Ser90] J.-P. Serre, Spécialisation des éléments de Br2(Q(T1,...,Tn)), C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 7, 397–402. A.10 Lucchini Arteche, Giancarlo If we make the term “higher-dimensional varieties” mean something like dimension greater or equal than, say, 10, then there aren’t too many families of varieties that are suitable for studying its rational points. A natural family of varieties that transcends in a certain sense this dimension problem is that of homogeneous spaces, that is, varieties admitting a transitive action of an algebraic group. For these varieties, the algebraic structure and the behavior of the rational points of the group itself can give us lots of information about the rational points of the homogeneous space, in particular (but not uniquely) via the use of the (usually non-abelian) Galois cohomology of these algebraic groups. Thus, if one is concerned for example with the problem of the Hasse principle (HP) or with weak approximation (WA), one can usually translate this statements into Galois cohomology statements, many of which have been proved in the last 50 years. In this trend, the deepest result up to date is probably that of Borovoi, who proved in 1996 that the Brauer-Manin obstruction to HP and WA is the only obstruction for homogeneous spaces under linear algebraic groups (over a number field) with connected or abelian stabiliser (the second case needing an aditional technical condition). This clearly leaves the open question: what about disconnected stabilisers ? In particular, for “simplicity”, what about finite (non-abelian) stabilisers ? These questions seem to be of a completely different nature of those already solved (compare for instance, over an algebraically closed field, the classification of connected reductive groups with that of finite groups !). However, if one believes in Colliot-Thlne’s conjecture on the Brauer-Manin obstruction to HP and WA being the only obstruction for unirational varieties, of which homogeneous spaces are a particular example, this questions should have the same answer. This is the problem I’m interested in today but, whether one cares about HP and WA or, for instance, integral points, strong approximation, problems in positive characteristic or any other arithmetico-geometric topic, homogeneous spaces have all the right to be studied in this workshop. A.11 Peyre, Emmanuel Currently I am interested in the notion of freeness for rational points which seems directly related to the characterization of accumulating subsets which appear in the study of rational points of bounded height on varieties. This notion is inspired by the notion of very free curves on varieties. A.12 Salberger, Per I am interested in the asymptotic behavior of the number of rational and integral points of bounded height. This interest started with the Manin conjectures. But as these could only be proved for special classes of varieties, I got interested in finding good upper estimates for varieties where the circle method give poor results like varieties of low dimension or very singular varieties. Most of my recent research has been devoted to refinements and 8 extensions of Heath-Browns p-adic determinant method, which often give better results than can be obtained by sieve theory. I gave in 2013 in Bonn a reformulation of the determinant method in terms of Mumfords geometric invariant theory (GIT), inspired by the use of GIT in Donaldsons and Tians work on Kähler-Einstein metrics. I have this year found a relation between the determinant method and generalisations of the Schmidt subspace theorem by Faltings-Wstholz and others. The aim of my present research is to obtain a deeper understanding of the determinant method by means of these links to other areas of mathematics. I have also recently collaborated with other mathematicians on the Manin conjectures. A.13 Sofos, Efthymios I am interested in Manin’s conjecture for smooth varieties in dimensions 2 and 3, especially in families for which current methods fail to provide the conjectured asymptotic, or even upper bounds of the correct order of magnitude. For these cases a lower bound of the correct order of magnitude is also not known, the only exception being Slater and Swinnerton-Dyer’s 1998 result which applies to smooth dP3 with two rational skew lines. In [1] we have developed an approach based on conic fibrations to give a straightforward proof of the Slater Swinnerton-Dyer result and to prove the analogous result for the Fermat cubic surface which is not covered by their theorem. It would be desirable to see whether this method can be used to cover larger families of smooth del Pezzo surfaces of degree ¡4 or smooth cubic 3-folds. [1] http://www.maths.bris.ac.uk/∼maxes/fermat.pdf A.14 Testa, Damiano There are several examples of diophantine equations that have been studied for many years before a connection with moduli spaces was noticed and exploited. Usually, finding such a connection, provides useful insights for studying arithmetic questions related to the diophantine problem. At the same time, it also explains previously observed geometric structures and uncovers new ones. Often, these connections are noticed by computing automorphism groups and Picard groups, or by finding special subvarieties and quotients. One of my interests in participating in the workshop is to explore possible connections between special diophantine equations and moduli spaces, learning about new examples, sharing the examples I already know and looking for new ones. A.15 Tschinkel, Yuri As one of the organizers of this workshop I am interested in all of the recent work of invited participants. I am curious about progress on the Brauer-Manin obstruction com- putations, about density results for rational and integral points, and about new geometric questions inspired by arithmetic. A.16 Wittenberg, Olivier I am interested in all aspects of rational and integral points of surfaces and higher- dimensional varieties, especially cubic and K3 surfaces. Recently I have worked on rational points and zero-cycles in fibrations. In joint work with Yonatan Harpaz and Alexei Skorobogatov (arXiv:1304.3333) we have shown that the results of Green, Tao and Ziegler yield a general theorem on the existence of rational points 9

1 on varieties fibered over PQ into varieties which satisfy weak approximation (under several serious hypotheses). In a work in progress with Yonatan Harpaz we formulate a conjecture on locally split values of polynomials which would imply an optimal result for rational points in fibrations (for reasonable general fibers, e.g. rationally connected). The particular case covering arXiv:1304.3333 follows from additive combinatorics (the results of Green-Tao- Ziegler), other particular cases can be proved by sieve methods (Irving). I would be very interested to learn what analytic methods have to say about this conjecture in general. A.17 Wooley, Trevor My interests lie in analytic methods for establishing existence of rational and integral points, usually on varieties of rather high dimension (not just exceeding two, but even exceeding the degree!). My favoured approach is the Hardy-Littlewood (circle) method, and I prefer my degrees to be large: at least 3 and usually exceeding 3. Here are some problems that need some new ideas:

Consider a hypersurface defined by the equation F (x1,...,xs) = 0, where F is homogeneous of degree d in s>d variables. Birch showed that when this hypersurface is non-singular and s > (d − 1)2d, then the circle method delivers an asymptotic formula for the number of integral solutions in a large box. There has been no progress (in 52 years) for general F in replacing this exponential growth in d by polynomial growth. (i) Is it possible to find a model for this hypersurface that has useful structure for analytic methods? E.g. quadratic polynomials can be diagonalised. Maybe there are sim- plifications one can make in some generality that are useful. (ii) Are there large new families of polynomials that can be successfully analysed an- alytically? Diagonal polynomials are perfect for the circle method, and splitting into many pieces with disjoint variables is almost as good. What about generalising linear forms in d d xi to quadratic or higher degree forms in xi ? Or polynomials with some kind of controlled perturbation from diagonal? Another area of activity focuses on the convexity barrier. Estimates for exponential sums of larger degree are currently limited by the square-root cancellation barrier. This limits applications typically to situations in which the number of variables exceeds twice the sum of degrees. There are a few situations in which one ducks inside this barrier: varieties for which the rational points are described by torsors simple enough to analyse completely, and pairs of diagonal cubics in 11 variables having a block structure of special type, for example. Can one devise other families of special varieties with similar properties? A.18 Yasufuku, Yu Because of my interest in Vojta’s conjecture, I am naturally drawn to rational and integral points on higher-dimensional varieties. Vojta’s conjecture is often brought up in cases when the canonical divisor is big (and so rational points are expected to be sparse), and this is done for a good reason, but I think there are very interesting and perhaps(?) provable cases of Vojta’s conjecture when the anti-canonical is ample (one of course needs to choose the simple normal-crossings divisor in an appropriate way). Since many of the participants of this workshop have studied and proved significant results in the direction of Manin conjecture, I would like to learn more details about various approaches and techniques that are used on these types of varieties.