In mathematics, a Hilbert modular surface is one of the surfaces obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group.

Hilbert modular surfaces were first described by using some unpublished notes written by Hilbert about 10 years before.

Definitions

If R is the ring of integers of a real quadratic field, then
the Hilbert modular group SL₂(R) 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:

• The surface X is the quotient of H×H by SL₂(R); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups.
• The surface X^* is obtained from X by adding a finite number of points corresponding to the cusps of the action. It is compact, and has not only the quotient singularities of X, but also singularities at its cusps.
• The surface Y is obtained from .X^* by resolving the singularities in a minimal way. It is a compact smooth algebraic surface, but is not in general minimal.
• The surface Y⁰ is obtained from Y by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal.

There are several variations of this construction:

• The Hilbert modular group may be replaced by some subgroup of finite index, such as a congruence subgroup.
• One can extend the Hilbert modular group by a group of order 2, acting on the Hilbert modular group via the Galois action, and exchanging the two copies of the upper half plane.

Singularities

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 surface

Rational surface

In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...

s or blown up K3 surface

K3 surface

In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....

s or elliptic surface

Elliptic surface

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....

s.

Examples

gives a long table of examples.

The Clebsch surface

Clebsch surface

In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface is a cubic surface studied by and all of whose 27 exceptional linescan be defined over the real numbers...

blown up at its 10 Eckardt points is a Hilbert modular surface.