Number theory seminar

Lance Gurney (U Amsterdam). Arithmetic of the moduli space of CM elliptic curves

In this talk I will describe certain arithmetic structures possessed by the moduli space M of elliptic curves with CM by the ring of integers O_K of a fixed imaginary quadratic field K. In the first part of the talk I will explain how M admits a commuting family of endomorphisms \psi_P: M --> M lifting the various Frobenius endomorphisms modulo each prime P of O_K. Exploiting this structure leads to generalisations of results of Gross on the existence of global minimal models and of Taylor-Cassou-Nogues on the monogenicity of the rings of integers of the ray class fields of K.

In the second part of the talk I will explain how the existence of the endomorphisms \psi_P is a consequence of the fact that M admits a Lambda-O_K-structure (a notion due to Borger). This implies the existence of a global, multi-prime theory of canonical lifts for CM elliptic curves analogous to the usual theory of Serre-Tate for ordinary elliptic curves. This allows for two curious constructions. The first is an infinite dimensional, smooth and affine cover of M and the second is an exact sequence related to both p-adic and complex periods.