Projective spaces contain lots of genus 1 curves. Does every twisted form of projective space contain a genus 1 curve? This basic arithmetic question was asked by David Saltman and Pete Clark and answered in low dimensions by Johan de Jong and Wei Ho using geometric methods. I will explain the history of the problem as well as recent joint work with Asher Auel where we show using arithmetic geometry, and the help of an observation of David Saltman, that the answer is `yes’ in small dimensions over global fields.