Brauer-Manin obstruction, K3 surfaces and families of twists of abelian varieties
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
Diophantine equations are one of the oldest parts of pure mathematics and the starting point in the development of number theory. Legendre and Gauss initiated a local-to-global approach to Diophantine equations culminating in class field theory and the Minkowski-Hasse theorem for quadratic forms over number fields. In essence this is the question about the passage from polynomial congruences modulo natural numbers to solutions of polynomial equations in integres. In 1970 Manin found a way to apply class field theory to the problem of existence of rational points on arbitrary algebraic varieties over global fields. The resulting theory of Brauer-Manin obstruction has had very many applications. It was later merged with the method of descent going back to Fermat, Mordell, Selmer, Cassels, and with the method of fibration going back to Hasse. These methods can be used to show that the Brauer-Manin obstruction controls the existence and distribution of rational points on certain geometrically rational varieties. As is traditional in number theory, the success of an algebraic technique depends on results from analytic number theory. Very strong analytic results have recently been obtained by Green, Tao and Ziegler by methods of additive combinatorics. As an application, important particular cases of long standing conjectures about rational families of conics and quadrics have been settled. On the other hand, for families of conics and quadrics parameterised by a curve of genus at least one, counterexamples have been found. K3 surfaces is athe next crucial class of algebraic varieties that in some sense occupies the middle ground between rational varieties, where one expects the behaviour of rational points to be controlled by the Brauer-Manin obstruction, and more general varieties where no efficient local-to-global approach is known. From another perspective, K3 surfaces are geometrically simply connected 2-dimensional analogues of elliptic curves, so one expects a deep and rich arithmetic theory of K3 surfaces and rational points on them. The only method to prove the existence of rational points on K3 surfaces known today is due to Swinnerton-Dyer. It applies to families of quadratic or cubic twists of abelian varieties, e.g. elliptic curves. The theory of elliptic curves has recently seen massive breakthroughs (due to Bharagava and others), and we hope to be able to use these results to advance our understanding of rational points on K3 surfaces and more general varieties.
Planned Impact
This project is fundamental research. I will benefit the London, British and international mathematical communities. The interdisciplinary nature of this research will help to promote communication between different fields of mathematics. Our results will lead to broader economic and social impact mainly through other scientists and academic users, as well as through bringing fresh talent into the highly successful UK university sector.
Direct applications outside of academia include applications in cryptography and security theory. Elliptic curves previously led to incredible advances in these areas since N.Koblitz and V.Miller introduced a form of public key cryptography based on their arithmetic. They are used in primality testing algorithms, and generally provide an indispensable tool in cryptography. These applications have become increasingly efficient as the relationships between the algorithmic and abstract aspects have become better understood. I have no doubt that the proposed research into the arithmetic geometry of rational points will lead to important developments in the long term, ultimately benefiting both governmental and private branches of the information security industry.
The results of this research can be of interest to communication theory and electrical and electronic engineering through the theory of error-correcting codes, also known as Goppa codes. Rational points on algebraic varieties are used when designing coding and decoding algorithms for algebraic-geometric codes constructed from these varieties, as well as for determining the parameters of these codes such as the number of correctable errors and the transmission rate. We hope to create tools that could be used later by those who apply rational points on algebraic varieties to create codes with better error-correcting properties.
Direct applications outside of academia include applications in cryptography and security theory. Elliptic curves previously led to incredible advances in these areas since N.Koblitz and V.Miller introduced a form of public key cryptography based on their arithmetic. They are used in primality testing algorithms, and generally provide an indispensable tool in cryptography. These applications have become increasingly efficient as the relationships between the algorithmic and abstract aspects have become better understood. I have no doubt that the proposed research into the arithmetic geometry of rational points will lead to important developments in the long term, ultimately benefiting both governmental and private branches of the information security industry.
The results of this research can be of interest to communication theory and electrical and electronic engineering through the theory of error-correcting codes, also known as Goppa codes. Rational points on algebraic varieties are used when designing coding and decoding algorithms for algebraic-geometric codes constructed from these varieties, as well as for determining the parameters of these codes such as the number of correctable errors and the transmission rate. We hope to create tools that could be used later by those who apply rational points on algebraic varieties to create codes with better error-correcting properties.
People |
ORCID iD |
Alexei Skorobogatov (Principal Investigator) |
Publications
Arne SMEETS
(2020)
Pseudo-split fibers and arithmetic surjectivity
in Annales scientifiques de l'École normale supérieure
Creutz B
(2018)
Degree and the Brauer-Manin obstruction
in Algebra & Number Theory
Daw Christopher
(2021)
Unlikely intersections with
E x CM curves in
A
2
in ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE
Gvirtz D
(2022)
Cohomology and the Brauer groups of diagonal surfaces
in Duke Mathematical Journal
Harpaz Y
(2016)
Hasse principle for Kummer varieties
in Algebra & Number Theory
Ieronymou E
(2017)
Corrigendum to "Odd order Brauer-Manin obstruction on diagonal quartic surfaces" [Adv. Math. 270 (2015) 181-205]
in Advances in Mathematics
Orr M
(2021)
On uniformity conjectures for abelian varieties and K3 surfaces
in American Journal of Mathematics
Orr M
(2022)
Invariant Brauer group of an abelian variety
in Israel Journal of Mathematics
Orr M
(2017)
On compatibility between isogenies and polarizations of abelian varieties
in International Journal of Number Theory
Orr M
(2018)
Finiteness theorems for K3 surfaces and abelian varieties of CM type
in Compositio Mathematica
Orr M
(2020)
Unlikely intersections with Hecke translates of a special subvariety
in Journal of the European Mathematical Society
Orr Martin
(2017)
Unlikely intersections with Hecke translates of a special subvariety
in arXiv e-prints
Orr Martin
(2021)
ON UNIFORMITY CONJECTURES FOR ABELIAN VARIETIES AND K3 SURFACES
in AMERICAN JOURNAL OF MATHEMATICS
Orr Martin
(2016)
Height bounds and the Siegel property
in arXiv e-prints
Orr Martin
(2018)
On uniformity conjectures for abelian varieties and K3 surfaces
in arXiv e-prints
Orr Martin
(2017)
Finiteness theorems for K3 surfaces and abelian varieties of CM type
in arXiv e-prints
Skorobogatov A
(2017)
Kummer varieties and their Brauer groups
in Pure and Applied Mathematics Quarterly
Skorobogatov A
(2015)
A Finiteness Theorem for the Brauer Group of K3 Surfaces in Odd Characteristic
in International Mathematics Research Notices
SKOROBOGATOV A
(2022)
ENRIQUES INVOLUTIONS AND BRAUER CLASSES
in Nagoya Mathematical Journal
Skorobogatov Alexei N.
(2016)
Kummer varieties and their Brauer groups
in arXiv e-prints
Description | Jointly with Yonatan Harpaz we proved the Hasse principle for Kummer varieties associated to abelian varieties that are sufficiently generic, conditionally on the finiteness of relevant Shafarevich-Tate groups. This gives a local-to-global principle for rational points on many K3 surfaces. Jointly with Martin Orr we proved a finiteness theorem for K3 surfaces of CM type defined over number fields of bounded degree. As a consequence we confirmed conjectures of Shafarevich and Varilly-Alvarado in the CM case. Domenico Valloni produced an explicit theory of complex multiplication on K3 surfaces, which allows one to calculate the transcendental Brauer group of CM K3 surfaces in many cases of interest. Otto Overcamp has proved the existence of Kulikov models for Kummer K3 surfaces. Alexei Skorobogatov in a joint work with Yuri Zarhin proved that the odd torsion subgroup of (generalised) Kummer varieties does not obstruct the Hasse principle. The same holds for the full Brauer-Manin obstruction in the case of Kummer varieties attached to hyperelliptic curves with large Galois action on 2-torsion. Further work has been done by Martin Orr and Alexei Skorobogatov on the inter-relations among the conjectures of Coleman, Shafarevich and Varilly-Alvarado for abelian varieties and K3 surfaces (joint work in preparation with Yuri Zarhin, Pennsylavnia State University). We proved that Coleman's conjecture implies the other two. |
Exploitation Route | Our results with Martin Orr and Yuri Zarhin set a path to further exploration of the arithmetic of K3 surfaces. We hope that our general results will encourage others to follow in our steps and do more concrete work for specific classes of K3 surfaces. |
Sectors | Digital/Communication/Information Technologies (including Software),Other |
URL | http://wwwf.imperial.ac.uk/~anskor/publ.htm |
Description | GLN |
Organisation | University of Bath |
Department | Department of Mathematical Sciences |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I brought in expertise on transcendental Brauer groups. |
Collaborator Contribution | They brought in expertise on algerbaic Brauer groups. |
Impact | Quantitative arithmetic of diagonal degree 2 K3 surfaces (arXiv:1910.06257) |
Start Year | 2018 |