One of the most exciting developments in number theory in the last decade is the revitalization of the geometry of numbers by Manjul Bhargava. By introducing novel new ideas, Bhargava and his co-authors proved many exciting theorems, including an unconditional upper bound for the average magnitude of the size of the Mordell-Weil group among elliptic curves defined over Q. The key to Bhargava’s methods is to count certain arithmetically interesting objects up to an appropriate equivalence relation. In this course we will give an introduction to the methods of Bhargava by focusing on counting binary forms. As a specific example, we show how counting binary quartic forms allows us to show that the average of the rank of the Mordell-Weil group of elliptic curves is at most 1.5.
Dr Stanley Xiao
These are the topics I intend to cover, but subject to change depending on time restrictions.
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Counting binary quadratic forms - classical results of Gauss and Siegel
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Counting binary cubic forms, and the Davenport-Heilbronn theorem
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Counting binary abelian binary cubic forms
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Counting binary quartic forms
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Binary quartic forms and 2-Selmer elements of elliptic curves
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Boundedness of the average rank of elliptic curves
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Counting Binary quartic forms whose Galois groups do not have elements
of order 3