Unexpected quadratic points on random hyperelliptic curves

Abstract: 

On a hyperelliptic curve over QQ, there are infinitely many points defined over quadratic fields: just pull back rational points of the projective line through the degree two map.  But for a positive proportion of genus g odd hyperelliptic curves over QQ, we give a bound on the number of quadratic points not arising in this way.  The proof uses tropical geometry work of Park, as well as that of Bhargava and Gross on average ranks of hyperelliptic Jacobians.  This is joint work with Jackson Morrow.