associated Hilbert modular variety obtained from compactifying a certain quotient variety and resolving its singularities. Let denote the upper half plane and let act on viawhere the are the Galois conjugates. The associated quotient variety is denotedand can be compactified to a variety , called the cusps, which are in bijection with the ideal classes in . Resolving its singularities gives the variety called the Hilbert modular variety of the field extension. From the Bailey-Borel compactification theorem, there is an embedding of this surface into a projective space. See also Hilbert modular form Picard modular surface Siegel modular variety References External links Algebraic surfaces Complex surfaces