Hilbert modular surface

In mathematics, a Hilbert modular surface or Hilbert–Blumenthal 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Hilbert modular surfaces were first described by Otto Blumenthal (1903, 1904) using some unpublished notes written by Hilbert about 10 years before.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:

There are several variations of this construction:

Singularities

Hirzebruch (1953) showed how to resolve the quotient singularities, and (Hirzebruch 1971) showed how to resolve their cusp singularities.

Classification of surfaces

The papers (Hirzebruch 1971), (Hirzebruch & Van der Ven 1974) and (Hirzebruch & Zagier 1977) 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

van der Geer (1988) gives a long table of examples.

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

See also

References

External links