Details of Grant 

EPSRC Reference:	EP/J005290/1
Title:	Arithmetic of Elliptic Curves and Abelian Varieties over Function Fields
Principal Investigator:	Pal, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	Imperial College London
Scheme:	Standard Research
Starts:	01 October 2012	Ends:	30 September 2015	Value (£):	342,469
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 30 Jan 2012 Mathematics Prioritisation Panel Meeting January 2012 Announced 05 Sep 2011 Mathematics Prioritisation Panel Meeting September 2011 Deferred
Summary on Grant Application Form
The study of elliptic surfaces over finite fields, or equivalently, of elliptic curves defined over function fields of transcendence degree one over finite fields is a venerable and highly developed area of arithmetic geometry. It is important testing area for central conjectures in the subject, such as the Tate, Beilinson and Parshin conjectures, but is also a rich source for discovering new phenomena. One of the major research directions of the area to arise in the last two decades is the study of these elliptic curves with the help of modular parameterizations by Drinfeld modular curves. Such parameterizations were successfully used to show special cases of the conjectures above, or prove other, more unexpected results. These developments required, and hence lead to a much finer understanding of these modular parameterizations, starting with the foundational work of Gekeler and Reversat. Not surprisingly the study of these parameterizations is now a very important subject of its own, with its own central problems. In this project we aim to resolve two central outstanding conjectures in the theory of modular parameterizations of elliptic curves by Drinfeld modular curves, namely Mazur's conjecture on the modular height of strong Weil curves, and the Gekeler-Reversat conjecture on the rigid analytic theta-functions of these modular parameterizations. We also expect that the methods which we will develop in order to resolve these decades-old conjectures will lead to the resolution of the uniform boundedness conjecture of Poonen-Schweizer, too.
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Organisation Website:	http://www.imperial.ac.uk