Non-CM Hilbert Modular Forms of Partial Weight One

Hilbert modular forms are a generalization of classical modular forms from the rational numbers to a totally real extension. Instead of a having a single weight, Hilbert modular forms have a vector of weights with one weight corresponding to each embedding of the field into the real numbers. We say a Hilbert modular form is of partial weight one if one, but not all, of its weights is equal to one. These forms are not well understood with the exception of those with CM, or complex multiplication. In fact, the first and only known example of a non-CM partial weight one form was only recently discovered. We will investigate partial weight one Hilbert modular forms and prove in a few cases that only CM forms exist.