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My proposed research is in number theory, an area of pure mathematics which is concerned with prime numbers and putative solutions to equations. It has long been understood in the field that there are surprisingly close analogies between number theory and the geometry of curves. The gradual arithmetic reinterpretation of geometric methods has been a continual source of inspiration and success for number theorists.However, in some cases it is the geometry of surfaces, not curves, which may provide the more powerful tools to study arithmetic questions; such questions belong to the realm of `two-dimensional arithmetic geometry'. Although the geometry of surfaces is now well-understood, there has been little success in transferring some of the more significant results from the geometric to the arithmetic setting. This project aims to investigate some of these problems using new techniques which have already been successful when reinterpreting the geometry of curves, and will hopefully help to build the two-dimensional geometric-to-arithmetic bridge.Objects of particular importance which are within the scope of this study are `elliptic curves', which are central objects of number theory, geometry, and cryptography. They played a fundamental role in A. Wiles' (et al.) celebrated proof in the 1990's of a conjecture posed by Fermat in the 17th century, which had resisted attack for 300 years. Further, elliptic curves are at the heart of one of the Clay Mathematics Institute's Millennium Problems, which are worth a prize of $1,000,000 to the solver.The research fellowship is to be held at Imperial College, London, because of the strong research group in arithmetic geometry spread between Imperial, University College, and King's College London, as well as the ease with which one can travel to the Universities of Cambridge, Oxford, and Nottingham. In order both to promote successful research and discuss future directions with experts in the relevant fields, there will be scientific trips abroad to the Hebrew University of Jerusalem, the University of Chicago, and Harvard University.More technically, this projects aims to study arithmetic surfaces adelically; that is, through the use of two-dimensional local fields and associated adelic spaces.The duality of such schemes will be investigated by first interpreting Grothendieck duality adelically, then attempting to twist Grothendieck duality by Poisson summation on the base, and finally seeking a reinterpretation of l-adic duality. Arithmetic intersection theory and arithmetic Riemann-Roch on the surface will be studied, and results compared with those arising from Arakelov theory.Relationships between conductors of curves over local fields and the ramification theory of two-dimensional local fields will be sought, starting with the case of an elliptic curve because of the availability of a cohomology-free definition of the conductor, via the Galois action on the Tate module. In the general case, comparison theorems between adelic invariants and l-adic cohomology will be required, and possible applications to the search for a two-dimensional Grothendieck-Ogg-Shavarevich formula will be considered.Finally, relationships between the geometric, arithmetic, and two-dimensional Langlands programmes will be studied, particularly through integration on loop groups.
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