{{Short description|Algebraic surface in mathematics}} 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modular surfaces were first described by {{harvs|txt|last=Blumenthal|first=Otto |year1=1903|year2=1904}} using some unpublished notes written by David Hilbert about 10 years before.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 SL<sub>2</sub>(''R'') acts on the product ''H''&times;''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''&times;''H'' by SL<sub>2</sub>(''R''); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups. *The surface ''X''<sup>*</sup> 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''<sup>*</sup> by resolving the singularities in a minimal way. It is a compact smooth algebraic surface, but is not in general minimal. *The surface ''Y''<sup>0</sup> is obtained from ''Y'' by blowing down certain exceptional &minus;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== {{harvtxt|Hirzebruch|1953}} showed how to resolve the quotient singularities, and {{harvtxt|Hirzebruch|1971}} showed how to resolve their cusp singularities.

== Properties == Hilbert modular varieties cannot be anabelian.<ref>{{cite book |last=Ihara |first=Yasutaka |author-link=Yasutaka Ihara |title=Geometric Galois Actions 1: Around Grothendieck's Esquisse d'un Programme |last2=Nakamura |first2=Hiroaki |editor-last1 = Schneps |editor-first1 = Leila |editor-link1 = Leila Schneps |editor-last2 = Lochak |editor-first2 = Pierre |editor-link2 = Pierre Lochak |publisher=Cambridge University Press |year=1997 |series=London Mathematical Society Lecture Note Series (242) |volume= |pages=127-138 |chapter=Some illustrative examples for anabelian geometry in high dimensions |doi=10.1017/CBO9780511758874.010}}</ref>

== Classification of surfaces == The papers {{harvtxt|Hirzebruch|1971}}, {{harvtxt|Hirzebruch|Van de Ven|1974}} and {{harvtxt|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== {{harvtxt|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.

=== Associated to a quadratic field extension === Given a quadratic field extension <math>K = \mathbb{Q}(\sqrt{p})</math> for <math>p = 4k + 1</math> there is an associated Hilbert modular variety <math>Y(p)</math> obtained from compactifying a certain quotient variety <math>X(p)</math> and resolving its singularities. Let <math>\mathfrak{H}</math> denote the upper half plane and let <math>SL(2,\mathcal{O}_K)/\{\pm \text{Id}_2\}</math> act on <math>\mathfrak{H}\times \mathfrak{H}</math> via<blockquote><math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} (z_1,z_2) = \left( \frac{ az_1 + bz_2 }{ cz_1 + dz_2 }, \frac{ a'z_1 + b'z_2 }{ c'z_1 + d'z_2 }\right) </math></blockquote>where the <math>a',b',c',d'</math> are the Galois conjugates.<ref>{{Cite book|last1=Barth|first1=Wolf P.|title=Compact Complex Surfaces|last2=Hulek|first2=Klaus|last3=Peters|first3=Chris A. M.|last4=Ven|first4=Antonius|date=2004|publisher=Springer Berlin Heidelberg|isbn=978-3-540-00832-3|location=Berlin, Heidelberg|pages=231|doi=10.1007/978-3-642-57739-0}}</ref> The associated quotient variety is denoted<blockquote><math>X(p) = G\backslash \mathfrak{H}\times\mathfrak{H}</math></blockquote>and can be compactified to a variety <math>\overline{X}(p)</math>, called the '''cusps''', which are in bijection with the ideal classes in <math>\text{Cl}(\mathcal{O}_K)</math>. Resolving its singularities gives the variety <math>Y(p)</math> 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.<ref>{{Cite journal|last1=Baily|first1=W. L.|last2=Borel|first2=A.|date=November 1966|title=Compactification of Arithmetic Quotients of Bounded Symmetric Domains|journal=The Annals of Mathematics|volume=84|issue=3|pages=442|doi=10.2307/1970457|jstor=1970457}}</ref>

==See also==

* Hilbert modular form * Picard modular surface * Siegel modular variety

==References== {{Reflist}} *{{Citation | last1=Barth | first1=Wolf P. |author-link=Wolf Barth | last2=Hulek | first2=Klaus |author2-link=Klaus Hulek | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 |mr=2030225 | year=2004 | volume=4 | doi=10.1007/978-3-642-57739-0}} *{{Citation | author1-link=Otto Blumenthal | last1=Blumenthal | first1=Otto | title=Über Modulfunktionen von mehreren Veränderlichen | doi=10.1007/BF01444306 | year=1903 | journal=Mathematische Annalen | volume=56 | issue=4 | pages=509–548 | s2cid=122293576 | url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002258986&L=1 | access-date=2013-09-12 | archive-date=2016-03-04 | archive-url=https://web.archive.org/web/20160304213701/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002258986&L=1 | url-status=dead | url-access=subscription }} *{{Citation | last1=Blumenthal | first1=Otto | title=Über Modulfunktionen von mehreren Veränderlichen | doi=10.1007/BF01449486 | year=1904 | journal=Mathematische Annalen | volume=58 | issue=4 | pages=497–527 | s2cid=179178108 | url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0058&DMDID=DMDLOG_0039&L=1 | access-date=2013-09-12 | archive-date=2016-03-04 | archive-url=https://web.archive.org/web/20160304190705/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0058&DMDID=DMDLOG_0039&L=1 | url-status=dead | url-access=subscription }} *{{Citation | last1=Hirzebruch | first1=Friedrich | author1-link=Friedrich Hirzebruch | title=Über vierdimensionale RIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen | doi=10.1007/BF01343146 | mr=0062842 | year=1953 | journal=Mathematische Annalen | issn=0025-5831 | volume=126 | issue=1 | pages=1–22 | hdl=21.11116/0000-0004-3A47-C | s2cid=122862268 | url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0126&DMDID=DMDLOG_0004&L=1 | hdl-access=free | access-date=2013-09-12 | archive-date=2016-03-05 | archive-url=https://web.archive.org/web/20160305074107/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0126&DMDID=DMDLOG_0004&L=1 | url-status=dead }} *{{Citation | last1=Hirzebruch | first1=Friedrich | author1-link=Friedrich Hirzebruch | title=Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 396 | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math | isbn=978-3-540-05720-8 | doi=10.1007/BFb0058707 |mr=0417187 | year=1971 | volume= 244 | chapter=The Hilbert modular group, resolution of the singularities at the cusps and related problems | pages=275–288| url=http://www.numdam.org/item/SB_1970-1971__13__275_0/ | url-access=subscription }} *{{Citation | last1=Hirzebruch | first1=Friedrich E. P. | title=Hilbert modular surfaces | doi=10.5169/seals-46292 |mr=0393045 | year=1973 | journal=L'Enseignement Mathématique |series=IIe Série | issn=0013-8584 | volume=19 | pages=183–281}} *{{Citation | last1=Hirzebruch | first1=Friedrich | author1-link=Friedrich Hirzebruch | last2=Van de Ven | first2=Antonius | title=Hilbert modular surfaces and the classification of algebraic surfaces | doi=10.1007/BF01405200 |mr=0364262 | year=1974 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=23 | issue=1 | pages=1–29| hdl=21.11116/0000-0004-39A4-3 | s2cid=73577779 | url=http://edoc.mpg.de/608462 | type=Submitted manuscript | hdl-access=free }} *{{Citation | last1=Hirzebruch | first1=Friedrich | author1-link=Friedrich Hirzebruch | last2=Zagier | first2=Don | editor1-last=Baily | editor1-first=W. L. | editor2-last=Shioda. | editor2-first=T. | title=Complex analysis and algebraic geometry | publisher=Iwanami Shoten | location=Tokyo | isbn=978-0-521-09334-7 |mr=0480356 | year=1977 | chapter=Classification of Hilbert modular surfaces | pages=43–77}} *{{Citation | last1=van der Geer | first1=Gerard | title=Hilbert modular surfaces | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-17601-5 |mr=930101 | year=1988 | volume=16 | doi=10.1007/978-3-642-61553-5}}

==External links== *{{citation|url=http://www.math.wisc.edu/~thyang/math941/hilbert_hz.pdf|first=S.|last= Ehlen|title=A short introduction to Hilbert modular surfaces and Hirzebruch-Zagier cycles}}

Category:Algebraic surfaces Category:Complex surfaces