Infinitely many rank 2 hyperelliptic curves

Based on the recent result of Peter Koymans and Adam Morgan, who
proved that there are infinitely many nonisomorphic hyperelliptic curves of
any genus whose Jacobian has rank 1 over any number field, we prove the
same result for rank 2; however, our curves do not have simple Jacobians,
instead they factor into two rank 1 Jacobians. We also discuss results
about higher rank Jacobians of small genus curves. This is joint work with
Sun Woo Park.