Hurwitz surface

<h2 id="classification-by-genus">Classification by genus</h2>
<p>Only finitely many Hurwitz surfaces occur with each genus. The function 
  
    
      
        h
        (
        g
        )
      
    
    {\displaystyle h(g)}
  
 mapping the genus to the number of Hurwitz surfaces with that genus is unbounded, even though most of its values are zero. The sum
</p>

∑
          
            i
            =
            1
          
          
            ∞
          
        
        
          
            
              h
              (
              g
              )
            
            
              g
              
                s
              
            
          
        
      
    
    {\displaystyle \sum _{i=1}^{\infty }{\frac {h(g)}{g^{s}}}}

<p>converges for 
  
    
      
        s
        >
        1
        
          /
        
        3
      
    
    {\displaystyle s>1/3}
  
, implying in an approximate sense that the genus of the 
  
    
      
        n
      
    
    {\displaystyle n}
  
th Hurwitz surface grows at least as a cubic function of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 (Larsen 2001).
</p><p>The Hurwitz surface of least genus is the <a href="/facts/Klein_quartic/yr7cHz5r">Klein quartic</a> of genus 3, with automorphism group the <a href="/facts/Projective_special_linear_group/y7P9rMyR">projective special linear group</a> <a href="/facts/PSL(2,7)/NFvGtws7">PSL(2,7)</a>, of order 84(3 − 1) = 168 = 23·3·7, which is a <a href="/facts/Simple_group/e5hGF7nT">simple group</a>; (or order 336 if one allows orientation-reversing isometries).  The next possible genus is 7, possessed by the <a href="/facts/Macbeath_surface/zqflzc4S">Macbeath surface</a>, with automorphism group PSL(2,8), which is the simple group of order 84(7 − 1) = 504 = 23·32·7; if one includes orientation-reversing isometries, the group is of order 1,008.
</p><p>An interesting phenomenon occurs in the next possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the identical automorphism group (of order 84(14 − 1) = 1092 = 22·3·7·13). The explanation for this phenomenon is arithmetic.  Namely, in the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of the appropriate <a href="/facts/Number_field/e0K0TXSh">number field</a>, the rational prime 13 splits as a product of three distinct <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a>.  The <a href="/facts/Principal_congruence_subgroup/V334rHwz">principal congruence subgroups</a> defined by the triplet of primes produce <a href="/facts/Fuchsian_group/qNrHQeUq">Fuchsian groups</a> corresponding to the <a href="/facts/First_Hurwitz_triplet/wzEdcttS">first Hurwitz triplet</a>.
</p><p>The sequence of allowable values for the genus of a Hurwitz surface begins
</p>
3, 7, 14, 17, 118, 129, 146, 385, 411, 474, 687, 769, 1009, 1025, 1459, 1537, 2091, ... (sequence A179982 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>)
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hurwitz_quaternion_order/S9ryNPyn">Hurwitz quaternion order</a></li></ul>

<ul><li>Elkies, N.: Shimura curve computations.  <i>Algorithmic number theory</i> (Portland, OR, 1998), 1–47, Lecture Notes in Computer Science, 1423, Springer, Berlin, 1998.  See <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.NT/0005160">math.NT/0005160</a></li>
<li>Hurwitz, A. (1893). "Über algebraische Gebilde mit Eindeutigen Transformationen in sich". <i><a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a></i>. 41 (3): 403–442. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01443420">10.1007/BF01443420</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122202414">122202414</a>.</li>
<li><a href="/facts/Mikhail_Katz/2Fy17Yyy">Katz, M.</a>; Schaps, M.; Vishne, U.: Logarithmic growth of <a href="/facts/Systolic_geometry/exKrWG9e">systole</a> of arithmetic Riemann surfaces along congruence subgroups.  J. Differential Geom. 76 (2007), no. 3, 399-422.  Available at <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.DG/0505007">math.DG/0505007</a></li>
<li>Larsen, Michael (2001). <a href="https://link.springer.com/article/10.1007/BF02784148"><i>How often is 84(g−1) achieved?</i></a>.</li>
<li>Singerman, David; Syddall, Robert I. (2003). <a href="http://www.emis.de/journals/BAG/vol.44/no.2/9.html">"The Riemann Surface of a Uniform Dessin"</a>. <i>Beiträge zur Algebra und Geometrie</i>. 44 (2): 413–430.</li></ul>

Hurwitz surface open-in-new

Hurwitz surface