K3 surfaces with high Neron-Severi rank -- Abhinav Kumar

I will talk about algebraic K3 surfaces (over $\mathbb{C}$) whose Neron-Severi lattice has high rank, in particular those K3 surfaces $X$ which have a Shioda-Inose structure, that is, $X$ has an involution $\iota$ which fixes any regular $(2,0)$-form, and the quotient $X/{1,\iota}$ is birational to a Kummer surface.

In particular, we can analyze moduli spaces of K3 surfaces with Shioda-Inose structure and a given polarization and its connection to the moduli spaces of polarized Kummer surfaces or of polarized abelian surfaces. In some cases, we can exhibit such K3 surfaces of high rank as elliptic surfaces with prescribed singular fibres, and here it is possible to explicitly write down an isogeny to a Kummer surface, following a method of Elkies. I will describe the computational details involved.