Hypergeometric local systems that carry a rational variation of Hodge structure are known to come from geometry. There are natural candidates for where they come from, the so-called canonical models, but their dimension is usually larger than the weight of the variation. In this talk I will look at the simplest interesting case, Hodge numbers (1,1), and explain how the canonical models can be cut down to elliptic surfaces. I’ll also describe all the integral structures sitting inside the rational variation and show where each one comes from geometrically. This is joint work in progress with Martin Mereb and Fernando Rodriguez Villegas.