Rational points on primary Burniat surfaces

Primary Burniat surfaces are certain surfaces of general type that
can be constructed as certain covers of a del Pezzo surface of degree 6, but
also as \'etale quotients of a surface contained in the product of three
elliptic curves. Using results by Chevalley-Weil and Faltings, the latter
description implies that rational points on primary Burniat surfaces are not
Zariski dense. We make this statement more concrete by determining the set of
low-genus curves on our surfaces. In favorable cases the set of rational
points can be determined explicitly; we will present some examples.
This is joint work with Ingrid Bauer.