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Hilbert modular surfaces were first described by Otto Blumenthal (1903, 1904) using some unpublished notes written by David Hilbert about 10 years before.
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: Hirzebruch (1953) showed how to resolve the quotient singularities, and Hirzebruch (1971) showed how to resolve their cusp singularities.
Hilbert modular varieties cannot be anabelian.
[1] The papers Hirzebruch (1971), Hirzebruch & Van de 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.
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.
Given a quadratic field extension
there is an associated Hilbert modular variety
obtained from compactifying a certain quotient variety
and resolving its singularities.
denote the upper half plane and let
{\displaystyle SL(2,{\mathcal {O}}_{K})/\{\pm {\text{Id}}_{2}\}}
are the Galois conjugates.
[2] 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.