# Rational points on varieties

### This project is no longer listed on FindAPhD.com and may not be available.

Dr Daniel Loughran  No more applications being accepted  Awaiting Funding Decision/Possible External Funding

A Diophantine equation is a polynomial equation where one seeks solutions in the integers or the rational numbers. Modern research mathematicians study such problems through the guise of rational points on varieties, in order to emphasise the geometric nature of the problem.

Basic questions are: is there a rational point? Are there infinitely many rational points? Can one obtain a finer quantitative description of the distribution of rational points? This project will make progress on understanding such problems for various classes of varieties (Fano varieties, norm equations, conic bundles, K3 surfaces).

To check whether a variety has a rational point, one first checks whether there is a real point and a p-adic point for all primes p. If this criterion is sufficient one says that the Hasse principle holds. In general the Hasse principle can fail, and this project will study such failures using the Brauer-Manin obstruction.

As for distribution of rational points, there is a conjecture of Manin which predicts the precise asymptotic behaviour of the number of rational points of bounded height. The project will try to make progress on this conjecture in some special cases.

In a similar philosophical vein, we could study the distribution of number fields of bounded discriminant which satisfy certain properties of algebraic interest. For example, studying the number of number fields of bounded discriminant which admit a normal integral basis. This complements Bhargava's work on counting number fields, for which he was awarded the Fields medal.

Depending on the interests and background of the student, the project could involve a range of techniques from analytic number theory, algebraic number theory, and algebraic geometry.

Early applications are encouraged as there may be an option to make an offer before the official closing date.

Candidate Requirements:

Applicants should hold, or expect to receive, a First Class or high Upper Second Class UK Honours degree (or the equivalent qualification gained outside the UK) in a relevant subject. A master’s level qualification would also be advantageous. The applicant should have an interest in number theory and algebraic geometry.

Non-UK applicants will also be required to have met the English language requirements of the University of Bath.

Enquiries and Applications:

Information enquiries are encouraged and should be directed to Dr Daniel Loughran on email address [Email Address Removed].

Formal applications should be made via the University of Bath's online application form for a PhD in Mathematics.

We welcome and encourage student applications from under-represented groups. We value a diverse research environment. If you have circumstances that you feel we should be aware of that have affected your educational attainment, then please feel free to tell us about it in your application form. The best way to do this is a short paragraph at the end of your personal statement.

## Funding Notes

it is anticipated that studentship funding will be obtained for this project covering tuition fees, a stipend at the UKRI standard doctoral rate (£15,609 p/a in 2021/22) and a training support allowance of £1,000 p/a for up to 4 years. This advertisement will be updated when funding in confirmed. In the meantime, interested candidates are advised to contact the lead supervisor for further information.

## References

Poonen, Bjorn. Rational points on varieties. Graduate Studies in Mathematics, 186. American Mathematical Society, Providence, RI, 2017.
Browning, Timothy D. Quantitative arithmetic of projective varieties. Progress in Mathematics, 277. Birkhäuser Verlag, Basel, 2009.

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