Diophantine geometry via analytic number theory
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
Polynomial equations are extremely commonplace in nature, and can be used to describe a myriad of physical and mathematical phenomena. For example, the theorem of Pythagoras states that a right-angled triangle with sides of lengths a<=b<=c has the property that these lengths always satisfy the quadratic equation a^2+b^2=c^2. It is a very natural step to try and determine under what circumstances a given polynomial equation admits integer solutions. For the Pythagorean equation this was answered completely by Diophantus in 250 AD, from whom we have inherited the term 'Diophantine equations'. Diophantus actually managed to write down the general solution in integers to Pythagoras' equation. For a general Diophantine equation, there are 3 basic possibilities: either we can show that there are infinitely many solutions (as above), or we can show that there are only finitely many solutions (as Wiles famously did for Fermat's equation a^k+b^k=c^k, when k>2), or we have trouble showing anything at all! The propensity for the third outcome lies at the heart of the enduring appeal that the subject of Diophantine equations enjoys.The equations mentioned so far have only involved 3 variables, and these are the equations that have been most closely studied. By contrast the solubility in integers of equations in 4 or more variables is still an untamed frontier, with only a scattering of results and conjectures on the map. One of the major outcomes of this project will be that many of the conjectural waypoints become established fact. The tools that I will use are rooted in analytic number theory, but will also take advantage of methods from algebraic geometry and the theory of descent. There is a useful interplay between Diophantine equations and the underlying geometry of the equation. This sort of connection provides a very useful source of extra leverage, and often reveals quite beautiful relations. My research makes essential use of this point of view.
Organisations
People |
ORCID iD |
Tim Browning (Principal Investigator) |
Publications
Baier S
(2012)
Inhomogeneous quadratic congruences
in Functiones et Approximatio Commentarii Mathematici
Baier S
(2012)
Averages of shifted convolutions of d 3 ( n )
in Proceedings of the Edinburgh Mathematical Society
Baier S
(2013)
Inhomogeneous cubic congruences and rational points on del Pezzo surfaces
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Browning T
(2011)
Power-free values of polynomials
in Archiv der Mathematik
Browning T
(2012)
Quadratic polynomials represented by norm forms
in Geometric and Functional Analysis
Browning T
(2011)
Least zero of a cubic form
in Mathematische Annalen
Browning T
(2009)
Manin's Conjecture for a Cubic Surface with D5 Singularity
in International Mathematics Research Notices
Browning T
(2011)
The divisor problem for binary cubic forms
in Journal de Théorie des Nombres de Bordeaux
Browning T
(2013)
Rational points on singular intersections of quadrics
in Compositio Mathematica
Browning T
(2009)
Linear growth for Châtelet surfaces
in Mathematische Annalen
Description | The Manin conjecture is established for several classes of geometrically rational varieties, including singular cubic surfaces and a family of Chatelet surfaces for which weak approximation fails. The Hasse principle is also established for some cubic hypersurfaces in dimension at least 11 and some normic equations. |
Exploitation Route | Nil |
Sectors | Other |
URL | http://www.maths.bris.ac.uk/~matdb/research.html |
Description | Frontiers of analytic number theory and selected topics |
Amount | £699,362 (GBP) |
Funding ID | ERC-2012-StG 306457-FANTAST |
Organisation | European Commission |
Sector | Public |
Country | European Union (EU) |
Start | 10/2012 |
End | 10/2017 |