Bo-Hae Im, A Hyperelliptic Curve Mapping to Specified Elliptic Curves

(This is a part of Arithemetic Geometry Day in IBS-CCG.)

We are interested in the existence and non-existence of rational curves on certain Kummer varieties which can be applied to the rank problem of quadratic twists of elliptic curves. In this talk, we prove that if the j-invariants of a 21-tuple of elliptic curves E₁, …, E₂₁ over C are algebraically independent over Q, there is no hyperelliptic curve which admits a non-trivial morphism to each of the E_i, which shows the non-existence of rational curves of the Kummer variety of the products of more than 20 elliptic curves.
This is a joint work with Michael Larsen and Sailun Zhan.