Details of Grant 

EPSRC Reference:	EP/D035619/1
Title:	The arithmetic of p-adic Hilbert modular forms
Principal Investigator:	Buzzard, Professor K
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Department:	Mathematics
Organisation:	Imperial College London
Scheme:	Standard Research (Pre-FEC)
Starts:	01 October 2005	Ends:	30 September 2008	Value (£):	78,473
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
In the 1970s, Langlands outlined a series of profound conjectures giving a strong relation between certain algebraic objects and certain analytic objects. Algebra and analysis are two main strands of modern pure mathematics, but they are sufficiently different that any link between them should be regarded as profound. These conjectures are known in a few cases but are mostly still wide open.Much more recently, people have been attempting to extend Langlands'picture to a link between certain p-adic algebraic and p-adic analyticobjects. Such links are becoming known as a p-adic Langlands programme . One case where one might hope to make progress is in the case of the group GL(2), where, for example, people have constructed p-adic families of 2-dimensional Galois representations and p-adic families of modular forms, and one can see the link between the two pictures in some cases.My proposal is to generalise these constructions to forms of GL(2) overtotally real fields; here the p-adic families of automorphic forms haveonly recently been constructed but enough is now known about them tobe optimistic that one can start proving instances of these links,for example, by constructing isomorphisms between various eigenvarieties coming from the various forms. The main difficulty is that the rigid geometry of Hilbert modular varieties (non-ordinary loci, canonical subgroups and so on) is much more poorly understood. However, very recent results in this area indicate that serious progress can now be made.
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Organisation Website:	http://www.imperial.ac.uk