  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.
