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10/04/05 Colloquium
Dr. Michael Joyce
Brown University
Brown University
Manin's Conjecture for the E_6 Cubic Surface
Abstract: Given a system of polynomial equations, number theorists are interested in studying solutions to this system whose coordinates are integers. A natural problem is to describe the asymptotic behavior of the number of solutions whose coordinates are all bounded in absolute value by B as B tends to infinity. Manin and others have conjectured that this asymptotic behavior is intimately related to geometric properties of the complex-valued soultions to the system of equations. We describe this conjecture and discuss how it has been verified for a specific highly singular cubic surface.