The moduli stack of elliptic curves (2019)
By moduli space of Riemann surfaces of genus g, where g is a non-negative integer, we mean the set of isomorphism classes of complex analytic structures on a closed oriented surface of genus g, fixed once and for all. It is not clear a priori why this definition makes sense, nor whether this set has an extra structure, turning it into some kind of space. In this memoir, we focus on these questions in the genus 1 case. As a matter of fact, we shall also fix some extra data: a base point on our genus 1 curve. The moduli space we obtain by doing this is called the moduli space of elliptic curves and the goal of the thesis is to show that this space is a complex analytic space, in a sense that we will make precise along the way.