Modular surfaces associated with toric K3 hypersurfaces

TL;DR

This paper explores the period maps of two families of toric K3 hypersurfaces, linking them to modular surfaces, and provides detailed descriptions of their geometric and modular properties.

Contribution

It introduces explicit descriptions of period maps for two specific families of toric K3 hypersurfaces, connecting them to modular surfaces.

Findings

One family related to a Hilbert modular surface.

Another related to a product of modular curves.

Provides detailed descriptions of the period maps.

Abstract

We give detailed descriptions of the period maps of two 2-parameter families of anti-canonical hypersurfaces in toric 3-folds. One of them is related to a Hilbert modular surface, and the other is related to the product of modular curves.