Periodic Reporting for period 1 - GIANT (Generalised Integrality and Applications to Number Theory)
Okres sprawozdawczy: 2024-06-01 do 2026-05-31
Podsumowanie kontekstu i ogólnych celów projektu
The main focus of GIANT is the study of semi-integral points. The notion of semi-integrality is a relatively new and little understood concept roughly corresponding to solutions of a system of polynomial equations in fractions with special denominators. On the contrary, studying solutions to polynomial equations dates back to the ancient Greeks at least, which is evident from the work of Diophantus of Alexandria. In modern days mathematicians look at the collection of all solutions to a system of polynomial equations as a geometric object. A single solution corresponds to a point on that object. Naturally, one is intrigued to study solutions in whole numbers (integer points) but it is often the case that the tools we have available are insufficient for understanding the structure of the underlying geometric object in order to extract information about integer points. Introducing denominators in a sense adds missing bits to the geometric object making it complete and thus easier to handle. Solutions in fractions (rational points) and how they are connected to the geometry of the underlying object have been much more understood, resulting in a rich theory for rational points. Restricting the allowed values of the denominators creates an intermediate notion of points in between the two classically studied ones described above. In the semi-integrality concept such restriction is controlled by a parameter. By varying the parameter a less or a more strict restriction on the denominators is applied. At the two extremes corresponding to the ends of the range of the spectrum of the parameter one recovers the two classical notions of rational and integral points.
Developing a theory for semi-integral points on one hand, creates a uniform theory for rational and integral points. On the other hand, extracting information about semi-integral points leads to better understanding of how integral points behave with respect to the arithmetic of underlying geometric object. For example, the lack of semi-integral points implies that no integral points exist. Currently, two different notions of semi-integral points depending on the types of conditions on the denominators are collectively abbreviated as semi-integral, they were first introduced by Campana (2004) and by Darmon (1997). Campana points have recently risen to the attention of the number theory community thanks to a Manin type conjecture, which predicts the total number of points in a bounded region, in the recent work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado (2021). Darmon points naturally correspond to questions about solutions of equations in perfect powers.
This project dives deeply in the following two directions (work packages WP1 & WP2) by exploring semi-integral points in families. The main goal of WP1 is to understand the quantity of geometric objects in general families with (or without) semi-integral points. The amount of objects in a family with rational points is predicted by the Loughran--Smeets conjecture but there is no general such conjecture for integral points. Only specific examples of families are studied when it comes to the lack of rational/integral points. In WP2 semi-integral points are studied in families where little to none is known for the qualitative or the quantitative behaviour of integral points. Of particular interest are families of geometric objects corresponding to long-standing open problems in number theory.
Developing a theory for semi-integral points on one hand, creates a uniform theory for rational and integral points. On the other hand, extracting information about semi-integral points leads to better understanding of how integral points behave with respect to the arithmetic of underlying geometric object. For example, the lack of semi-integral points implies that no integral points exist. Currently, two different notions of semi-integral points depending on the types of conditions on the denominators are collectively abbreviated as semi-integral, they were first introduced by Campana (2004) and by Darmon (1997). Campana points have recently risen to the attention of the number theory community thanks to a Manin type conjecture, which predicts the total number of points in a bounded region, in the recent work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado (2021). Darmon points naturally correspond to questions about solutions of equations in perfect powers.
This project dives deeply in the following two directions (work packages WP1 & WP2) by exploring semi-integral points in families. The main goal of WP1 is to understand the quantity of geometric objects in general families with (or without) semi-integral points. The amount of objects in a family with rational points is predicted by the Loughran--Smeets conjecture but there is no general such conjecture for integral points. Only specific examples of families are studied when it comes to the lack of rational/integral points. In WP2 semi-integral points are studied in families where little to none is known for the qualitative or the quantitative behaviour of integral points. Of particular interest are families of geometric objects corresponding to long-standing open problems in number theory.
Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty
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.
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.
Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu)
The research performed both in WP1 and WP2 introduces new approaches and results that directly exceed the current state of art. Uematsu showed that the traditional approach of employing the Brauer group globally does not work for the two families of diagonal surfaces in WP2, therefore the counting results obtained in WP2 would have been impossible without a new input. In fact, previously it was not even clear what the expectation for the correct order of magnitude of the counting problems studied in WP2 should have been.
On the other hand, GIANT completes only the initial part of a program aimed at the quantitative study of semi-integral points in families that requires further research. The local-global principle for Brauer elements developed in WP1 is not restricted to specific families of orbifold pairs or algebraic varieties. This opens the door to a wide range of possible applications. The hyperbola method of WP2 is of independent interest to the number theory community and can be used to study counting problems beyond the applications of this project, e.g. Manin's conjecture and Loughran--Smeets conjecture.
On the other hand, GIANT completes only the initial part of a program aimed at the quantitative study of semi-integral points in families that requires further research. The local-global principle for Brauer elements developed in WP1 is not restricted to specific families of orbifold pairs or algebraic varieties. This opens the door to a wide range of possible applications. The hyperbola method of WP2 is of independent interest to the number theory community and can be used to study counting problems beyond the applications of this project, e.g. Manin's conjecture and Loughran--Smeets conjecture.