Arithmetic of hyperelliptic curves

Hyperelliptic curves are a fundamental class of polynomial equations that has featured in geometry and number theory for a very long time, but whose arithmetic has not yet been subject to a systematic study. This gap in our knowledge is rapidly becoming apparent, as demands for the theory are coming both from within pure mathematics, from areas bordering to theoretical physics (via the new theory of hypergeometric motives), and from cryptography, where one of the main methods for modern data encryption is based on hyperelliptic curves.

The purpose of the project is to modernise our approach to the arithmetic of hyperelliptic curves, by bringing in the number theoretic machinery of L-functions and Selmer groups into the subject. These are tools that have been the centre of attention of many number theorists over the past few decades, and lie at the heart of the works on Fermat's Last Theorem, the Langlands programme and the Birch-Swinnerton-Dyer conjecture. They promise to provide new theoretical and computational techniques for working with hyperelliptic curves, and our aim is to establish foundational results whose analogues have been central to the development of other parts of number theory.

From the point of view of the established theory of L-functions, the step into hyperelliptic curves is partly a step into unchartered territory, for hyperelliptic curves cannot be treated by the comfortably familiar techniques based on modular forms. We thus plan to expand and test the L-function theory beyond its standard boundaries, and hope to shed light on the many unresolved conjectures in the subject.

The interplay of hyperelliptic curves, L-functions and Selmer groups is the rationale for proposing a single unified project. Our aim is to produce mathematical results, algorithms and data that can be used in each of these three worlds. Apart from establishing results for number theorists, we aim to explore phenomena and develop concrete classifications and a database, that would also be accessible to scientists from other fields working with hyperelliptic curves.

** Who will benefit? **

a) Knowledge Economy

b) Mathematicians working in number theory, arithmetic and algebraic geometry, Galois representations, L-functions, Iwasawa theory and the Langlands programme; in the long term, cryptographers, both in academia and in industry, and physicists

c) UK universities, especially Bristol and Warwick, and our academic partners

d) UK economy in general


** How will they benefit? **

Knowledge economy. The project concerns fundamental research in the one activity that underpins all scientific endeavour (whether a physical, medical or a social science), namely mathematics. It is set up to answer some questions on the forefront of the modern mathematical research, and it will increase our understanding of some of the most basic concepts - numbers and equations - that humans operate with. The circle of questions underlying the project has led to famous fundamental advances in the 20th century (see Academic Beneficiaries), and we are confident that the project itself and the data it generates will serve as a backbone for similarly fundamental breakthroughs in the 21st century mathematics.

Mathematicians. Breakthroughs in mathematics often come through intradisciplinary links, connections between research areas. The proposed project links several of them - arithmetic and algebraic geometry, Galois representations, L-functions, Iwasawa theory and the Langlands programme, and connects theoretical research to algorithmic questions and numerical experiments. Researchers in these areas will benefit from having these links and from having a large collection of data that slices through the areas. Cryptographers, especially those working in hyperelliptic curves, exploring their limits and looking for new directions will also benefit from having access to this data, as will physicists who study links with L-functions and hypergeometric motives. In the long term, this offers additional societal benefits, especially through applications to cryptography, which is vital for the security and reliability of the internet, digital commerce, and national security.

UK universities will benefit through the raised international profile of their mathematics research, both through the attraction of top young international researchers and through the links to the world centres of excellence, such as Princeton, Sydney and ICTP Trieste.

The UK economy will benefit from the increased number of skilled mathematicians who are not only experts in their specialised field, but who have learned to work within several research areas, and mastered a wide variety of theoretical, computational and data-management techniques. Whether they choose to continue in academia or contribute to digital and knowledge economy, they will be highly sought after.