Details of Grant 

EPSRC Reference:	EP/G026378/1
Title:	Congruences of Siegel modular forms
Principal Investigator:	Dummigan, Dr NP
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Researcher Co-Investigators:
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Department:	Pure Mathematics
Organisation:	University of Sheffield
Scheme:	Overseas Travel Grants (OTGS)
Starts:	12 January 2009	Ends:	11 April 2009	Value (£):	7,055
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
Galois groups were invented to capture the symmetries of the roots of polynomialequations, for example to show that the quadratic formula does not generaliseto equations of degree 5 or higher. Polynomial equations in more than one variable lead to algebraic geometry, and when the equations have rational coefficients, one naturally acquires representationsof Galois groups as symmetries of spaces (with any number of dimensions),in which the coordinates of points are not ordinary numbers, but belong tonumber systems derived from modular arithmetic.Complex numbers were introduced in order to be able to solve all quadraticequations, but are also often the natural kind of numbers to use for calculusand geometry. A modular form is a kind of very symmetric function of complexvariables. Galois representations give rise to L-functions, a different type offunctions of a complex variable. L-functions generalise the Riemann zetafunction, famous for its application to the distribution of prime numbers, andfor the unsolved Riemann hypothesis.There are deep connections, mostly conjectural, between modular forms,Galois representations and values of L-functions. Such theoretical phenomenacan sometimes have down-to-earth arithmetical consequences, for examplethe proof of Fermat's Last Theorem. I propose to study some congruences between modular forms which, through the associated Galois representations, have implications for values of L-functions.
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Organisation Website:	http://www.shef.ac.uk