Hilbert modular variety


In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group.
Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before.

Definitions

If R is the ring of integers of a real quadratic field, then
the Hilbert modular group SL2 acts on the product H×H of two copies of the upper half plane H.
There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces:
There are several variations of this construction:
showed how to resolve the quotient singularities, and showed how to resolve their cusp singularities.

Classification of surfaces

The papers , and identified their type in the classification of algebraic surfaces. Most of them are surfaces of general type, but several are rational surfaces or blown up K3 surfaces or elliptic surfaces.

Examples

gives a long table of examples.
The Clebsch surface blown up at its 10 Eckardt points is a Hilbert modular surface.

Associated to a quadratic field extension

Given a quadratic field extension for there is an associated Hilbert modular variety obtained from compactifying a certain quotient variety and resolving it's singularities. Let denote the upper half plane and let act on via
where the are the Galois conjugates. The associated quotient variety is denoted
and 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.