On the structure of mixed weight Hilbert modular forms

In this talk we discuss joint work with Sho Takemori on Hilbert modular forms
over the real quadratic field of discriminant 5, with respect to its full modular group.
The graded ring of all Hilbert modular forms of parallel weight were determined by Gundlach.
By using his result and some elementary technique, we establish a structure theorem on mixed
Hilbert modular forms.