elliptic k3 surface and Shioda Inose structure

Question

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$?

For example, in the note http://www2.iag.uni-hannover.de/~schuett/K3-fam.pdf , in section 10, the authors seem could recover the pair of elliptic curves according to the elliptic fibration. But I have no idea how do they make it. Is there any suggestion for that?

Thanks

Answer 1

Recovering the pair $(E, E')$ there are two aspects. Abstractly this is just a matter of Hodge theory, so theoretically it is easy. Explicitly, however, is a totally different story, and the only way that I am aware of, is to find one of the elliptic fibrations (with section) on $\text{Km}(E \times E')$ which were classified by Oguiso in 1989, see here.

• K. Oguiso. On Jacobian fibrations on the Kummer surfaces of the product of nonisogenous elliptic curves. J. Math. Soc. Japan 41, 651–680 (1989).

In fact, the one with fiber $IV^*$ twice or $II^*$ and $I^*_0$ twice allow you to read off the pair of $j$-invariants directly (this is what is used in the paper of Elkies and Schütt), following what Inose and Shioda did, see here.

• T. Shioda and H. Inose. On singular K3 surfaces. Complex analysis and algebraic geometry, pp. 119–136. Iwanami Shoten, Tokyo, 1977.

Inose has a single-authored paper with explicit formulas which already includes everything that is needed, see here.

• H. Inose. Defining equations of singular K3 surfaces and a notion of isogeny. Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), pp. 495–502, Kinokuniya Book Store, Tokyo, 1978.

The note of Elkies and Schütt is rather sketchy, but for instance, you could consult Schütt's paper with Garbagnati where similar computations are carried out explicitly, see here.

• A. Garbagnati and M. Schütt. Enriques surfaces: Brauer groups and Kummer structures. Michigan Math. J. 61, 297–330 (2012).