Details of Grant 

EPSRC Reference:	GR/S18519/01
Title:	Zeta Functions of groups and arithmetic geometry
Principal Investigator:	du Sautoy, Professor M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematical Institute
Organisation:	University of Oxford
Scheme:	Standard Research (Pre-FEC)
Starts:	01 July 2003	Ends:	30 September 2006	Value (£):	127,750
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Panel History:
Summary on Grant Application Form
The proposal lies in the boundary between group theory and number theory. Zeta functions of groups were introduced originally as potentially new invariants in attempts to understand the difficult problem of classifying infinite nilpotent subgroups. Recent work of the Principal Investigator has revealed that these zeta functions are an equally important weapon in trying to understand the problem of classifying the wild class of finite p-groups. The zeta functions of groups are a non-commutative analogue of the zeta function of a number field. The zeta function seeks to encode the subgroup structure of an infinite group. However, recent work of the Principal Investigator with Grunewald has shown that this analogy with the Dedekind zeta function of a number field is too simplistic. Rather it is the zeta function of Artin, Hasse and Weil counting points modulo p on varieties that offers the better model. The Principal Investigator subsequently constructed an example of a nilpotent group whose lattice of subgroups encodes the arithmetic an elliptic curve. This example was used as a counterexample to a question proposed at the birth of the theory of zeta functions. This proposal seeks investigate the new dialogue between arithmetic geometry and group theory that this example uncovered. In particular to explore what sort of other varieties can be realised in a similar fashion and what varieties arise in connection with the zeta functions counting finite p-groups. This will impact on the 40 year old PORC conjecture due to Higman.
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Organisation Website:	http://www.ox.ac.uk