ARITHMETIC MODULI OF GENERALIZED ELLIPTIC CURVES

Abstract

The theory of generalized elliptic curves gives a moduli-theoretic compactification for modular curves when the level is a unit on the base, and the theory of Drinfeld structures on elliptic curves provides moduli schemes over the integers without a modular interpretation of the cusps. To unify these viewpoints it is natural to consider Drinfeld structures on generalized elliptic curves, but some of these resulting moduli problems have non-etale automorphism groups and so cannot be Deligne–Mumford stacks. Artin’s method as used in the work of Deligne and Rapoport rests on a technique of passage to irreducible fibers (where the geometry determines the group theory), and this does not work in the presence of non-etale level structures and non-etale automorphism groups. By making more efficient use of the group theory to bypass these difficulties, we prove that the standard moduli problems for Drinfeld structures on generalized elliptic curves are proper Artin stacks. We also analyze the local structure on these stacks and give some applications to Hecke correspondences.