Equivariant Conjectures in Arithmetic

An important theme in modern number theory is the exploration of the relation between certain analytic objects on the one hand and certain algebraic objects that encode arithmetic information on the other hand. For example, the Birch and Swinnerton-Dyer conjecture relates the arithmetic of an elliptic curve over a number field to the behaviour of its Hasse-Weil L-function. It is widely recognized as one of the most challenging problems in mathematics; indeed, the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a million dollar prize for the first correct proof.

The leading term conjectures at s=0 and s=1 are formulated for arbitrary finite Galois extensions of number fields, and they relate leading terms of Artin L-functions to certain natural arithmetic invariants. These can be seen as generalisations and refinements of the Stark conjectures; in particular, they recover the analytic class number formula (up to sign). Moreover, the global and local epsilon constant conjectures assert that the two leading term conjectures are compatible in some sense. The proposed research consists of several innovative lines of attack on these conjectures.

This can be seen in a wider context as follows. When specialised to the case of Tate motives (i.e. Galois extensions of number fields), the equivariant Tamagawa number conjecture (ETNC) of Burns and Flach recovers the leading term conjectures (one has to assume Leopoldt's conjecture to recover the leading term conjecture at s=1). When the motive in question is an elliptic curve defined over Q, the ETNC recovers the Birch and Swinnerton-Dyer conjecture. Indeed, a key reason for interest in the ETNC is that it provides an elegant and unifying framework for conjectures involving leading terms of L-functions attached to motives. Moreover, this framework means that new results on the leading term conjectures should lead to important progress on the ETNC for more complicated motives.

The academic impact of the proposed research is described in the "Academic Beneficiaries" section above. Thus we address the non-academic impact here.

In the long term, research in pure mathematics can have unforeseen applications which are of wider benefit to society. Moreover, research in mathematics makes a substantial contribution to the UK economy; this is discussed at length in the EPSRC-commissioned report "Measuring the Economic Benefits of Mathematical Science Research in the UK" which is available at http://www.cms.ac.uk/files/Submissions/article_EconomicBenefits.pdf

For instance, number theory (the area in which the proposed research lies) has important applications to secure digital communication via coding theory and cryptography. Indeed, as stated in the executive summary of the above report "Carl Friedrich Gauss' work on number theory from the 18th and 19th centuries, previously thought to have little practical use, now underpins much modern work in cryptology, data management and the encoding of digital data."

Moreover, training in the mathematical sciences has a huge impact in terms of UK employment, especially in sectors such as finance and computer services. Indeed, in the "Quantifiable Impacts of MSR in 2010'' section of the above report, it is stated that in 2010, there were 2.8 million individuals employed in mathematical science occupations in the UK and the gross added value associated with mathematical science research was 208 billion pounds.

Thus it is important to share general mathematical skills and knowledge with both graduate and undergraduate students, as well as other researchers in mathematics. Exeter provides an excellent environment in which to interact with other mathematicians and students, and is in easy reach of other mathematically active universities such as Bristol, Bath, Oxford, Warwick and the universities in London, for example.

I also plan to participate in outreach activities that could include, for example, giving talks to sixth formers on popular topics in number theory such as Fermat's Last Theorem or cryptography, thereby encouraging students to think about the possibility of studying mathematics at university. The recently opened Exeter Mathematics School (jointly sponsored by the University of Exeter and Exeter College) offers ample opportunities for such activities.