Pencils of quadrics and the arithmetic of hyperelliptic curves

In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over Q of genus g have no points over any odd degree extension of Q. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser-Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory (the study of rational orbits for representations of reductive groups) and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages.