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The arithmetic of hyperelliptic curves

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The arithmetic of hyperelliptic curves

Manjul Bhargava has recently made significant progress on the arithmetic ofelliptic curves over Q. Together with his student Arul Shankar, he has calculated the averageorder of the n-Selmer group, for n = 2,3,4,5, and has obtained an upper bound on theaverage rank (which is less than one). To do this, they identify elements of the Selmer groupwith certain orbits in a representation of a semi-simple group over Q, and estimatethe number of orbits of bounded height using the geometry of numbers. In this talk, which is a report on joint work with Bhargava, I will explain how thesetechniques can be extended to study the arithmetic of hyperelliptic curves of a fixedgenus over Q, with a marked rational Weierstrass point.

- Variétés rationnelles