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
Browning T
(2011)
Least zero of a cubic form
in Mathematische Annalen
Browning, T D
(2009)
Manin's conjecture for a cubic Surface with D5 singularity
Browning, T.D. (Oxford University)
(2009)
Quantitative Arithmetic of Projective Varieties
De La Bretèche R
(2010)
Le problème des diviseurs pour des formes binaires de degré 4
in Journal für die reine und angewandte Mathematik (Crelles Journal)
De La Bretèche R
(2008)
Binary linear forms as sums of two squares
in Compositio Mathematica
De La Bretèche R
(2011)
Manin's conjecture for quartic Del Pezzo surfaces with a conic fibration
in Duke Mathematical Journal
De La Bretèche R
(2012)
Binary forms as sums of two squares and Châtelet surfaces
in Israel Journal of Mathematics
De La Bretèche R
(2012)
On Manin's conjecture for a family of Châtelet surfaces
in Annals of Mathematics
Matthiesen L
(2013)
Correlations of representation functions of binary quadratic forms
in Acta Arithmetica
Tim Browning (Author)
(2009)
Solubility of Fermat equations
in Quadratic Forms-Algebra, Arithmetic, and Geometry
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 |