Arithmetic of hyperelliptic curves

Lead Research Organisation: University of Bristol
Department Name: Mathematics

Abstract

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.

Planned Impact

** 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.

Publications

10 25 50

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Dokchitser T (2018) Quotients of hyperelliptic curves and Étale cohomology in The Quarterly Journal of Mathematics

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Dokchitser T (2018) 3-torsion and conductor of genus 2 curves in Mathematics of Computation

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Dokchitser T (2021) Tate module and bad reduction in Proceedings of the American Mathematical Society

 
Description 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 was 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 was to modernise our approach to the arithmetic of hyperelliptic curves, by bringing in the number theoretic machinery of Galois representations, 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 was to establish foundational results whose analogues have been central to the development of other parts of number theory.

We believe that this search for foundational results was successful. We were able to develop a `local' theory of hyperelliptic curves of arbitrary genus that is strong enough to analyze their arithmetic invariants. This collaborative effort is described in over 200 pages of published or submitted manuscripts, and will hopefully serve as a backbone for researchers interested both in the local and the global arithmetic of hyperelliptic curves. For example, it allows to work with the Birch-Swinnerton-Dyer conjecture for elliptic curves and carry out experiments of the kind that were not possible before, beyond elliptic curves. Along the way we created, and made publicly available, a database of hyperelliptic curves of small genus, that could be used as a base for these experiments. These results have already been used by several authors; and in particular, two research associates on the project - Bartosz Naskrecki and Celine Maistret relied on them to resolve previously unsolved questions on the structure of hypergeometric motives of low degree, and parity of 2-ranks of Jacobians.

A surprising fact that we discovered is that some of the techniques that came out of the project apply to arbitrary curves, not necessarily hyperelliptic. This is something that we are going to explore in the future, but the directions look very promising, and have attracted considerable interest internationally.
Exploitation Route Research in pure mathematics takes a long route to impact, so it is early to tell what the medium or long term impact of the project is going to be, though the initial feedback from the academic community has been very encouraging. For the short-term impact:
- Mathematicians will benefit from new techniques that we have developed and documented;
- UK and EU academic environment will profit greatly from the workshops and the summer school that we have organised; we received very positive feedback on that;
- Postdoctoral researchers on our team have developed an excellent set of skills and experience of working together as a team. A few of them went into research-heavy industry, some will stay in academia, and it appears that all of them have landed excellent jobs as a result of taking part in the project.
Sectors Digital/Communication/Information Technologies (including Software)

 
Title Hyperelliptic curves of small genus 
Description A database of hyperelliptic curves of small genus (2,3,4,5) and bounded discriminant; made publicly available in 2016. 
Type Of Material Database/Collection of data 
Year Produced 2016 
Provided To Others? Yes  
Impact N/A yet 
URL https://people.maths.bris.ac.uk/~matyd/HE/index.html
 
Description KCL 
Organisation King's College London
Country United Kingdom 
Sector Academic/University 
PI Contribution All work is joint between the two teams of researchers, one at Bristol and one at KCL (formerly Warwick)
Collaborator Contribution All work is joint between the two teams of researchers, one at Bristol and one at KCL (formerly Warwick)
Impact Work in progress
Start Year 2015
 
Title Software to compute regular models of curves 
Description Package to compute regular models of curves, based on the paper "Models of curves over DVRs", June 2018. 
Type Of Technology Software 
Year Produced 2018 
Open Source License? Yes  
Impact The software has been released recently, but it is already being used in research, as far as I am aware. 
URL https://people.maths.bris.ac.uk/~matyd/newton/
 
Description STEM Parliamentary Poster Competition 2018 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Policymakers/politicians
Results and Impact STEM Parliamentary Poster Competition 2018. Dr Celine Maistret presented her poster related to her research on hyperelliptic curves, and won the first prize (Gold Award) in Mathematics.
Year(s) Of Engagement Activity 2018