EPSRC Reference: 
EP/S004696/2 
Title: 
Localglobal principles: arithmetic statistics and obstructions 
Principal Investigator: 
Newton, Dr RD 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Kings College London 
Scheme: 
New Investigator Award 
Starts: 
01 August 2021 
Ends: 
30 September 2022 
Value (£): 
28,992

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Summary on Grant Application Form 
Methods for solving polynomial equations in integers and rationals have been sought and studied for more than 4000 years. Sometimes it is easy to see that a polynomial equation admits no 'global' (meaning integral or rational) solution. For example, if the equation has no solution in the real numbers, then it clearly has no integer solution. The reason that looking for real solutions is easier is because the real numbers have a desirable property called completeness, which relates to the fact that the real numbers form a continuum with no gaps. It is the discrete nature of the integers which makes them difficult to deal with. By viewing the integers within other exotic number systems (called the padic numbers) that also enjoy the property of completeness, we can avail ourselves of other ways to rule out existence of integer solutions to polynomial equations. If the equation has no padic solution then it has no integer solution. But what if the equation has solutions in the field of real numbers and in all the padic fields? Does this mean it has a rational solution? If we have a family of equations where the answer to this question is yes, then we say the Hasse principle holds for that family. For example, the Hasse principle holds for quadratic forms. This means that determining whether a quadratic form has an integer solution is easy. However, there are equations of degree 3 and higher for which the Hasse principle fails. This leads to some natural questions, such as: How often does the Hasse principle fail? Why does it fail?
This research addresses both of these questions for certain families of equations. To answer the first question, we will fix a family of equations which can be enumerated in a meaningful way. We will then determine whether the Hasse principle can fail for any equation in the family. For those equations where failures can occur, we will calculate an algebraic object which measures the severity of the failure and determines the precise local conditions which are responsible for the failure. The most difficult step will be to calculate what proportion of the equations in the family give failures. This will tell us whether failure is, as we hope, a rare occurrence in the family. If failures are rare, then a randomly chosen equation in the family will satisfy the Hasse principle and determining whether it has a global solution is equivalent to checking whether it has real and padic solutions. The latter calculation can be performed in finite time, whereas no general algorithm exists for determining whether a polynomial equation has an integer solution.
The second question concerns obstructions to localglobal principles such as the Hasse principle. The most important known obstruction is the BrauerManin obstruction. There are several challenges to be overcome in order to understand the consequences of the BrauerManin obstruction for a family of varieties. One must calculate the Brauer group, which is the algebraic object quantifying the obstruction. Then one must calculate the obstruction given by each element of the Brauer group. Finally, one must determine whether the BrauerManin obstruction suffices to explain all failures of localglobal principles in the family. The second part of this project will push the boundaries of our current understanding of each step in this process.

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