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Hilbert modular surfaces were first described by Otto Blumenthal (1903, 1904) using some unpublished notes written by Hilbert about 10 years before.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

• 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

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.

Picard modular surface

References

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
Blumenthal, Otto (1903), "Über Modulfunktionen von mehreren Veränderlichen", Mathematische Annalen (Berlin, New York: Springer-Verlag) 56 (4): 509–548, doi:10.1007/BF01444306
Blumenthal, Otto (1904), "Über Modulfunktionen von mehreren Veränderlichen", Mathematische Annalen (Berlin, New York: Springer-Verlag) 58 (4): 497–527, doi:10.1007/BF01449486
Hirzebruch, Friedrich (1953), "Über vierdimensionale RIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen", Mathematische Annalen 126 (1): 1–22, doi:10.1007/BF01343146, ISSN 0025-5831, MR 0062842
Hirzebruch, Friedrich (1971), "The Hilbert modular group, resolution of the singularities at the cusps and related problems", Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 396, Lecture Notes in Math 244, Berlin, New York: Springer-Verlag, pp. 275–288, doi:10.1007/BFb0058707, ISBN 978-3-540-05720-8, MR 0417187
Hirzebruch, Friedrich E. P. (1973), "Hilbert modular surfaces", L'Enseignement Mathématique. Revue Internationale. IIe Série 19: 183–281, doi:10.5169/seals-46292, ISSN 0013-8584, MR 0393045
Hirzebruch, Friedrich; Van de Ven, Antonius (1974), "Hilbert modular surfaces and the classification of algebraic surfaces", Inventiones Mathematicae 23 (1): 1–29, doi:10.1007/BF01405200, ISSN 0020-9910, MR 0364262
Hirzebruch, Friedrich; Zagier, Don (1977), "Classification of Hilbert modular surfaces", in Baily, W. L.; Shioda., T., Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 43–77, ISBN 978-0-521-09334-7, MR 0480356
van der Geer, Gerard (1988), Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 16, Berlin, New York: Springer-Verlag, ISBN 978-3-540-17601-5, MR 930101