Moduli of CM False Elliptic Curves

Abstract

We study two moduli problems involving false elliptic curves with complex multiplication (CM), generalizing theorems about the arithmetic degree of certain moduli spaces of CM elliptic curves. The first moduli problem generalizes a space considered by Howard and Yang, and the formula for its arithmetic degree can be seen as a calculation of the intersection multiplicity of two CM divisors on a Shimura curve. This formula is an extension of the Gross-Zagier theorem on singular moduli to certain Shimura curves. The second moduli problem we consider deals with special endomorphisms of false elliptic curves. The formula for its arithmetic degree generalizes a theorem of Kudla, Rapoport, and Yang.