Arithmetic moduli of generalized elliptic curves -- Brian Conrad

A generalized elliptic curve is what is obtained when an elliptic curve is allowed to degenerate in a controlled way that retains some group-theoretic information. There is a good theory of proper moduli stacks for generalized elliptic curves with "etale level structure" (due to Deligne and Rapoport) and also a good theory of non-proper moduli stacks for elliptic curves with possibly non-etale level structure (due to Katz and Mazur, building on ideas of Drinfeld). These stacks are all Deligne-Mumford. It is natural to ask if these theories can be "glued" to make a theory of proper moduli stacks for generalized elliptic curves with possibly non-etale level structure. I will explain the additional ideas that are needed to achieve this (after discussing some background and examples), and a concrete arithmetic application will be given. One notable feature is the intervention of Artin stacks that are not Deligne-Mumford.