Details of Grant 

EPSRC Reference:	EP/E049109/1
Title:	Two dimensional adelic analysis
Principal Investigator:	Fesenko, Professor I
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Sch of Mathematical Sciences
Organisation:	University of Nottingham
Scheme:	Standard Research
Starts:	01 December 2007	Ends:	28 February 2011	Value (£):	381,744
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
This research proposal is in number theory and its interaction with algebraic geometry, analysis, algebraic K-theory, topology, model theory. The subject of the research is the study of fundamental properties of zeta functions of elliptic curves over global fields and their regular minimal models using new analytical and arithmetical theory, methods and tools in geometrical setting, which form the two dimensional adelic analysis programme. Zeta functions is the most important object in number theory.The Langlands correspondence for zeta and L-functions (interelations of automorphicand Galois theoretical aspects of zeta functions), the Riemann hypothesis for zeta functions (location of their zeros or poles), the Birch and Swinnerton-Dyer conjecture (special behaviour of the zeta function at integer points) are major open and very difficult problems in number theory. The research of this proposal aims to study the fundamental issues of the zeta functionsof elliptic curves and regular models using new complex analytic methods, ideas and tools. This is a two dimensional extension of the fundamental work of Tate and Iwasawa. The research of this proposal will have applications to the meromorphic continuation and functional equation of the zeta and L-function of elliptic curves over global fields, the Riemann hypothesis for the zeta function, the Birch and Swinnerton-Dyer conjecture. It also aims to initiate and develop the theory of automorphic functions on arithmetic surfaces, in the central case of elliptic surfaces. The International Review of UK Mathematics indicates one direction that has played a major role outside the UK, and which seems never to have developed a critical mass in the UK is the modern theory of automorphic forms, for example, the Langlands programme. The subject and methods of the two dimensional adelic analysis are interrelated with various aspects of the Langlands correspondence in many different ways. The work on the project includes support of visits to the UK of several world leaders in the Langlands correspondence.
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Project URL:	http://www.maths.nott.ac.uk/personal/ibf/049109.pdf
Further Information:
Organisation Website:	http://www.nottingham.ac.uk