A long time ago, inspired by a paper of Mestre constructing genus 2 curves whose Jacobians had real multiplication by Q(sqrt(5)), I wrote a paper showing the existence of continuous families of curves X whose Jacobians had real multiplication by various abelian extensions of Q. I constructed these curves as branched covers with prescribed ramification, which is to say I had no real way of presenting them explicitly at all. I just saw a nice preprint by Ivan Boyer, a recent Ph.D. student of Mestre, which takes all the curves I construct and computes explicit equations for them! I wouldn’t have thought this was doable (in particular, I never thought about whether the families in my construction were rational.) For instance, for any value of the parameter s, the genus 3 curve
has real multiplication by the real subfield of . Cool!