Lance Gurney's Homepage

I am a post-doc working in the Algebra, Geometry and Mathematical Physics Group at the Universiteit van Amsterdam. Please send all emails to [firstname].[lastname]@gmail.com.

Research Interests:

Witt vectors, p-divisible groups, Lubin-Tate groups, class field theory, lifts of Frobenius, canonical lifts, elliptic curves with complex multiplication, Drinfel'd modules of rank one, Coleman functions.

Papers:

Canonical lifts of elliptic curves (with James Borger)

We construct functorial canonical lifts of arbitrary families of ordinary elliptic curves over schemes on which p is locally nilpotent.

Elliptic curves of Shimura type

A new characteristation of elliptic curves of Shimura type is given in terms of commuting families of Frobenius lifts. This combined with a strengthening of an old principal ideal theorem is applied to the existence of minimal models of such elliptic curves (generalising previous results of Gross).

Elliptic curves with complex multiplication and Λ-structures (Thesis)

The following is a brief overview, for more details please see the introduction. It includes a detailed study of the moduli stack of elliptic curves equipped with complex multiplication, the two new results being that it is a torsor under a stack of rank one local systems and that it admits a Λ-structure, this essentially means that arbitrary families of elliptic curves with complex multiplication admit lifts, or deformations, to schemes of big Witt vectors. These structures are then used to prove the existence of minimal models for elliptic curves of Shimura type, to construct an integral explicit class field theory for imaginary quadratic fields, to construct a rigidification of the moduli stack of elliptic curves with complex multiplication by an infinite dimensional Grassmannian and to construct an embedding of the Tate module of an elliptic curve with complex multiplication into the group of points of a certain deformation of the given elliptic curve.