Geometry and arithmetic of primary Burniat surfaces

We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as 'etale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus 1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a given primary Burniat surface $S$ defined over $\mathbb{Q}$. This involves an explicit description of the relevant twists of the 'etale covering of $S$ coming from the second construction mentioned above and methods for finding the set of rational points on a given twist.