Orbits of group representations, and arithmetic applications (I)

Manjul Bhargava, Princeton

We describe how various fundamental algebraic structures (involving, for example, number fields, class groups, and algebraic curves) can be parametrized via the orbits of appropriate group representations. By developing techniques to count such orbits, we can then use these parametrizations to prove analytic density theorems about the corresponding algebraic objects.

In these two talks and the colloquium, we discuss several recent such results (as well as the techniques behind them), and in particular, we lead up to a proof that the average rank of all elliptic curves (when ordered by their heights) is bounded. [In fact, it is bounded above by 1.05 .] As a consequence, we obtain that a positive proportion of elliptic curves satisfy the Birch and Swinnerton-Dyer conjecture.