Hilbert modular variety

<h2 id="definitions">Definitions</h2>
<p>If <i>R</i> is the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of a real <a href="/facts/Quadratic_field/dUnSDN4e">quadratic field</a>, then 
the Hilbert modular group SL2(<i>R</i>) <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acts</a> on the product <i>H</i>×<i>H</i> of two copies of the upper half plane <i>H</i>.
There are several <a href="/facts/Birational_geometry/hsWqRhJ1">birationally equivalent</a> surfaces related to this action, any of which may be called <a href="/facts/Hilbert_modular_surface/TmoIXYs8">Hilbert modular surfaces</a>:
</p>
<ul><li>The surface <i>X</i> is the quotient of <i>H</i>×<i>H</i> by SL2(<i>R</i>); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups.</li>
<li>The surface <i>X</i>* is obtained from <i>X</i> by adding a finite number of points corresponding to the <a href="/facts/Cusp_(singularity)/QH3fEGry">cusps</a> of the action. It is compact, and has not only the quotient singularities of <i>X</i>, but also singularities at its cusps.</li>
<li>The surface <i>Y</i> is obtained from <i>X</i>* by resolving the singularities in a minimal way. It is a compact smooth <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surface</a>, but is not in general minimal.</li>
<li>The surface <i>Y</i>0 is obtained from <i>Y</i> by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal.</li></ul>
<p>There are several variations of this construction:
</p>
<ul><li>The Hilbert modular group may be replaced by some subgroup of finite index, such as a <a href="/facts/Congruence_subgroup/V334rHwz">congruence subgroup</a>.</li>
<li>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.</li></ul>
<h2 id="singularities">Singularities</h2>
<p>Hirzebruch (1953) showed how to resolve the quotient singularities, and Hirzebruch (1971) showed how to  resolve their cusp singularities.
</p>
<h2 id="properties">Properties</h2>
<p>Hilbert modular varieties cannot be <a href="/facts/Anabelian/zqvz8gqf">anabelian</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="classification-of-surfaces">Classification of surfaces</h2>
<p>The papers  Hirzebruch (1971), Hirzebruch & Van de Ven (1974) and Hirzebruch & Zagier (1977) identified their type in the <a href="/facts/Classification_of_algebraic_surfaces/Lr1rd0Nf">classification of algebraic surfaces</a>. Most of them are <a href="/facts/Surfaces_of_general_type/GEkqj4Lm">surfaces of general type</a>, but several are <a href="/facts/Rational_surface/8LMRdT0m">rational surfaces</a> or blown up <a href="/facts/K3_surface/hqbR3jpu">K3 surfaces</a> or <a href="/facts/Elliptic_surface/XRDYGhYb">elliptic surfaces</a>.
</p>
<h2 id="examples">Examples</h2>
<p>van der Geer (1988) gives a long table of examples.
</p><p>The <a href="/facts/Clebsch_surface/qwDJptGE">Clebsch surface</a> blown up at its 10 Eckardt points is a Hilbert modular surface.
</p>
<h3>Associated to a quadratic field extension</h3><p>
Given a <a href="/facts/Quadratic_field/dUnSDN4e">quadratic field extension</a> 
  
    
      
        K
        =
        
          Q
        
        (
        
          
            p
          
        
        )
      
    
    {\displaystyle K=\mathbb {Q} ({\sqrt {p}})}
  
 for 
  
    
      
        p
        =
        4
        k
        +
        1
      
    
    {\displaystyle p=4k+1}
  
 there is an associated Hilbert modular variety 
  
    
      
        Y
        (
        p
        )
      
    
    {\displaystyle Y(p)}
  
 obtained from compactifying a certain quotient variety 
  
    
      
        X
        (
        p
        )
      
    
    {\displaystyle X(p)}
  
 and resolving its singularities. Let 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathfrak {H}}}
  
 denote the upper half plane and let 
  
    
      
        S
        L
        (
        2
        ,
        
          
            
              O
            
          
          
            K
          
        
        )
        
          /
        
        {
        ±
        
          
            Id
          
          
            2
          
        
        }
      
    
    {\displaystyle SL(2,{\mathcal {O}}_{K})/\{\pm {\text{Id}}_{2}\}}
  
 act on 
  
    
      
        
          
            H
          
        
        ×
        
          
            H
          
        
      
    
    {\displaystyle {\mathfrak {H}}\times {\mathfrak {H}}}
  
 via</p><p>
  
    
      
        
          
            (
            
              
                
                  a
                
                
                  b
                
              
              
                
                  c
                
                
                  d
                
              
            
            )
          
        
        (
        
          z
          
            1
          
        
        ,
        
          z
          
            2
          
        
        )
        =
        
          (
          
            
              
                
                  a
                  
                    z
                    
                      1
                    
                  
                  +
                  b
                  
                    z
                    
                      2
                    
                  
                
                
                  c
                  
                    z
                    
                      1
                    
                  
                  +
                  d
                  
                    z
                    
                      2
                    
                  
                
              
            
            ,
            
              
                
                  
                    a
                    ′
                  
                  
                    z
                    
                      1
                    
                  
                  +
                  
                    b
                    ′
                  
                  
                    z
                    
                      2
                    
                  
                
                
                  
                    c
                    ′
                  
                  
                    z
                    
                      1
                    
                  
                  +
                  
                    d
                    ′
                  
                  
                    z
                    
                      2
                    
                  
                
              
            
          
          )
        
      
    
    {\displaystyle {\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)}
  
</p><p>where the 
  
    
      
        
          a
          ′
        
        ,
        
          b
          ′
        
        ,
        
          c
          ′
        
        ,
        
          d
          ′
        
      
    
    {\displaystyle a',b',c',d'}
  
 are the <a href="/facts/Galois_conjugate/6jbUaXd3">Galois conjugates</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The associated quotient variety is denoted</p><p>
  
    
      
        X
        (
        p
        )
        =
        G
        ∖
        
          
            H
          
        
        ×
        
          
            H
          
        
      
    
    {\displaystyle X(p)=G\backslash {\mathfrak {H}}\times {\mathfrak {H}}}
  
</p><p>and can be compactified to a variety 
  
    
      
        
          
            X
            ¯
          
        
        (
        p
        )
      
    
    {\displaystyle {\overline {X}}(p)}
  
, called the cusps, which are in bijection with the <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal classes</a> in 
  
    
      
        
          Cl
        
        (
        
          
            
              O
            
          
          
            K
          
        
        )
      
    
    {\displaystyle {\text{Cl}}({\mathcal {O}}_{K})}
  
. Resolving its singularities gives the variety 
  
    
      
        Y
        (
        p
        )
      
    
    {\displaystyle Y(p)}
  
 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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hilbert_modular_form/BjIRAzCX">Hilbert modular form</a></li>
<li><a href="/facts/Picard_modular_surface/2jH9nlan">Picard modular surface</a></li>
<li><a href="/facts/Siegel_modular_variety/DAVwkHCI">Siegel modular variety</a></li></ul>

<ul><li>Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), <i>Compact Complex Surfaces</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-57739-0">10.1007/978-3-642-57739-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-00832-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2030225">2030225</a></li>
<li><a href="/facts/Otto_Blumenthal/25tvaTp5">Blumenthal, Otto</a> (1903), <a href="https://web.archive.org/web/20160304213701/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002258986&L=1">"Über Modulfunktionen von mehreren Veränderlichen"</a>, <i>Mathematische Annalen</i>, 56 (4): 509–548, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01444306">10.1007/BF01444306</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122293576">122293576</a>, archived from <a href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002258986&L=1">the original</a> on 2016-03-04, retrieved 2013-09-12</li>
<li>Blumenthal, Otto (1904), <a href="https://web.archive.org/web/20160304190705/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0058&DMDID=DMDLOG_0039&L=1">"Über Modulfunktionen von mehreren Veränderlichen"</a>, <i>Mathematische Annalen</i>, 58 (4): 497–527, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01449486">10.1007/BF01449486</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:179178108">179178108</a>, archived from <a href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0058&DMDID=DMDLOG_0039&L=1">the original</a> on 2016-03-04, retrieved 2013-09-12</li>
<li><a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Hirzebruch, Friedrich</a> (1953), <a href="https://web.archive.org/web/20160305074107/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0126&DMDID=DMDLOG_0004&L=1">"Über vierdimensionale RIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen"</a>, <i><a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a></i>, 126 (1): 1–22, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01343146">10.1007/BF01343146</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/21.11116%2F0000-0004-3A47-C">21.11116/0000-0004-3A47-C</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0062842">0062842</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122862268">122862268</a>, archived from <a href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0126&DMDID=DMDLOG_0004&L=1">the original</a> on 2016-03-05, retrieved 2013-09-12</li>
<li><a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Hirzebruch, Friedrich</a> (1971), "The Hilbert modular group, resolution of the singularities at the cusps and related problems", <a href="http://www.numdam.org/item/SB_1970-1971__13__275_0/"><i>Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 396</i></a>, Lecture Notes in Math, vol. 244, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. 275–288, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0058707">10.1007/BFb0058707</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05720-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0417187">0417187</a></li>
<li>Hirzebruch, Friedrich E. P. (1973), "Hilbert modular surfaces", <i>L'Enseignement Mathématique</i>, IIe Série, 19: 183–281, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5169%2Fseals-46292">10.5169/seals-46292</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0013-8584">0013-8584</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0393045">0393045</a></li>
<li><a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Hirzebruch, Friedrich</a>; Van de Ven, Antonius (1974), <a href="http://edoc.mpg.de/608462">"Hilbert modular surfaces and the classification of algebraic surfaces"</a>, <i><a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a></i> (Submitted manuscript), 23 (1): 1–29, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01405200">10.1007/BF01405200</a>, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/21.11116%2F0000-0004-39A4-3">21.11116/0000-0004-39A4-3</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0364262">0364262</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:73577779">73577779</a></li>
<li><a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Hirzebruch, Friedrich</a>; Zagier, Don (1977), "Classification of Hilbert modular surfaces", in Baily, W. L.; Shioda., T. (eds.), <i>Complex analysis and algebraic geometry</i>, Tokyo: Iwanami Shoten, pp. 43–77, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-09334-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0480356">0480356</a></li>
<li>van der Geer, Gerard (1988), <i>Hilbert modular surfaces</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-61553-5">10.1007/978-3-642-61553-5</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-17601-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0930101">0930101</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Ehlen, S., <a href="http://www.math.wisc.edu/~thyang/math941/hilbert_hz.pdf"><i>A short introduction to Hilbert modular surfaces and Hirzebruch-Zagier cycles</i></a> (PDF)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Ihara, Yasutaka; Nakamura, Hiroaki (1997). "Some illustrative examples for anabelian geometry in high dimensions". In Schneps, Leila; Lochak, Pierre (eds.). Geometric Galois Actions 1: Around Grothendieck's Esquisse d'un Programme. London Mathematical Society Lecture Note Series (242). Cambridge University Press. pp. 127–138. doi:10.1017/CBO9780511758874.010. <a href="/wiki/Yasutaka_Ihara" target="_blank">/wiki/Yasutaka_Ihara</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Ven, Antonius (2004). Compact Complex Surfaces. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 231. doi:10.1007/978-3-642-57739-0. ISBN 978-3-540-00832-3. <a href="978-3-540-00832-3" target="_blank">978-3-540-00832-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Baily, W. L.; Borel, A. (November 1966). "Compactification of Arithmetic Quotients of Bounded Symmetric Domains". The Annals of Mathematics. 84 (3): 442. doi:10.2307/1970457. JSTOR 1970457. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Hilbert modular variety open-in-new

Hilbert modular variety