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Geometry-of-numbers techniques and counting orbits of binary cubic forms

4:00 pm Thursday, April 10, 2014
Arul Shankar (Harvard)

I will discuss a classical result of Davenport, establishing asymptotics for the number of $Gl_2(Z)$-orbits on integral binary cubic forms having bounded discriminant, with a power saving error term. I will then discuss an averaging method of Bhargava's that improves Davenport's power saving error term to an optimal one. Finally, we introduce a refinement that allows us to determine the second main term for this count! The final refinement is joint work with Bhargava and Tsimerman.

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