Elliptic curves with complex multiplication and $\Lambda$-structures

TL;DR

This thesis explores the structure of elliptic curves with complex multiplication, demonstrating their Lambda structures, lifting properties to Witt vectors, and connections to reciprocity maps and period theories.

Contribution

It establishes that moduli stacks of CM elliptic curves admit Lambda structures and introduces new models and relationships in the theory of CM elliptic curves.

Findings

Moduli stacks of CM elliptic curves admit Lambda structures.

CM elliptic curves can be canonically lifted to Witt vectors.

A new rigidification of the moduli stack of CM elliptic curves is constructed.

Abstract

This thesis examines the relationship between elliptic curves with complex multiplication and Lambda structures. Our main result is to show that the moduli stack of elliptic curves with complex multiplication, and the universal elliptic curve with complex multiplication over it, both admit Lambda structures and that the structure morphism is a Lambda morphism. This implies that elliptic curves with complex multiplication can be canonically lifted to the Witt vectors of the base (these are big and global Witt vectors). We also show that elliptic curves with complex multiplication of Shimura type are precisely those admitting Lambda structures and that a large class of these elliptic curves admit global minimal models. Along the way, we present a detailed study of families of elliptic curves with complex multiplication over arbitrary bases, give new derivations of the reciprocity maps associated to local fields and imaginary quadratic fields, construct a new flat, affine and pro-smooth rigidification of the moduli stack of elliptic curves with complex multiplication and exhibit a relationship between perfect Lambda schemes and periods, both $p$-adic and analytic.