The geometric object corresponding to the collection of all solutions to a system of polynomial equations together with the data about the conditions imposed on the denominators form a pair, known as a Campana orbifold or an orbifold pair. The main goal of WP1 is to understand quantitatively the distribution of pairs in families with (or without) semi-integral points. An obvious reason why a pair may fail to have semi-integral points is the lack of such points locally. For example, if a polynomial equation has an integral solution, substituting that solution in and reducing the equation modulo some prime power yields a solution modulo that prime power. A local point at a given prime (place) consists of a class of solutions modulo powers of that prime, where roughly two solutions which are essentially the same but arise modulo the different prime powers have been identified. Every semi-integral (global) point gives rise to a local point at each place but not vice versa. If for a given family every orbifold pair that has local points at each place also has a global point, the family satisfies a local-global principle known as the Hasse principle.
It is a widely open and notoriously difficult question why members of a general family may fail the Hasse principle. However, failures of that principle for rational points are understood at least conjecturally for less complex families. By Colliot-Thélène's conjecture all such failures are explained by a construction known as the Brauer--Manin obstruction. In crude terms that obstruction uses all local points together with an algebraic object, namely the Brauer group of the corresponding member of the family, to create a subset of all local points that contains all global points. If that subset is empty, there are no global points and hence the Hasse principle fails. The main issue in understanding the Brauer--Manin obstruction in families is often the fact that elements of the Brauer group cannot be described uniformly across the family.
To deal with this issue, we have developed a new method as part of WP1, which identifies those orbifold pairs in the family that have a Brauer--Manin obstruction to the Hasse principle. Instead of understanding elements of the global Brauer group across the family, the method relies on understanding the local Brauer groups which are much more accessible and whose elements can be tracked across the family. Extracting the data from the behaviour of local Brauer elements and piecing it together essentially works a local-global principle for Brauer elements. In this method, all places are naturally divided in two categories: big (generic) and small (exceptional). Often the main difficulty lies in understanding what happens at small places. If suitable analytic tools are available for counting under the conditions output by the local-global principle for Brauer elements, this approach allows one to study the number of Hasse principle failures in the family and to obtain for this quantity:
- an asymptotic formula, if both small and big places can be handled;
- a sharp upper bound, if only big places can be analysed.
WP2 is devoted to understanding how the Brauer--Manin obstruction is distributed across families of paris of specific interest to the number theory community. One such family consists of orbifold pairs coming from Markoff surfaces where we performed an extensive study of the quantitative and qualitative behaviour of Darmon and for Campana points. In particular, we also studied other local-global principles for such pairs, namely strong and weak approximation.
In WP2 a generalised version of the hyperbola method has been developed. This is a new analytic tool that we combine with the method introduced in WP1 to study two families of orbifold pairs of interest to the number theory community. The first application is in the setting of integral points on diagonal affine quadric surfaces. It has been known for more than a century that such surfaces satisfy the Hasse principle for rational points. However, they may fail the Hasse principle for integral or semi-integral points. All such failures are explained by the Brauer--Manin obstruction. WP2 completely resolves the question which diagonal affine quadric surfaces fail to have integral points and how they are distributed across the family. The second application is motivated by a question first studied by Mordell in 1942, who was interested in cubic surfaces failing the Hasse principle for rational points. Subsequently, cubic surfaces with no rational points were studied by Manin in the 1970s and later by Colliot-Thélène, Kanevsky and Sansuc in 1987. For them Colliot-Thélène's conjecture predicts that all failures of the Hasse principle are explained by the Brauer--Manin obstruction. In WP2 we finally settle the question which diagonal cubic surfaces fail the Hasse principle and how often this happens.
