First Hurwitz triplet

<h2 id="arithmetic-construction">Arithmetic construction</h2>
<p>Let 
  
    
      
        K
      
    
    {\displaystyle K}
  
 be the real subfield of 
  
    
      
        
          Q
        
        [
        ρ
        ]
      
    
    {\displaystyle \mathbb {Q} [\rho ]}
  
 where 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
 is a 7th-primitive <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a>. 
The <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> of <i>K</i> is 
  
    
      
        
          Z
        
        [
        η
        ]
      
    
    {\displaystyle \mathbb {Z} [\eta ]}
  
, where 
  
    
      
        η
        =
        2
        cos
        ⁡
        (
        
          
            
              
                2
                π
              
              7
            
          
        
        )
      
    
    {\displaystyle \eta =2\cos({\tfrac {2\pi }{7}})}
  
. Let 
  
    
      
        D
      
    
    {\displaystyle D}
  
 be the <a href="/facts/Quaternion_algebra/CYtNzofE">quaternion algebra</a>, or symbol algebra 
  
    
      
        (
        η
        ,
        η
        
          )
          
            K
          
        
      
    
    {\displaystyle (\eta ,\eta )_{K}}
  
. Also Let 
  
    
      
        τ
        =
        1
        +
        η
        +
        
          η
          
            2
          
        
      
    
    {\displaystyle \tau =1+\eta +\eta ^{2}}
  
 and 
  
    
      
        
          j
          ′
        
        =
        
          
            
              1
              2
            
          
        
        (
        1
        +
        η
        i
        +
        τ
        j
        )
      
    
    {\displaystyle j'={\tfrac {1}{2}}(1+\eta i+\tau j)}
  
.  Let 
  
    
      
        
          
            
              Q
            
          
          
            
              H
              u
              r
            
          
        
        =
        
          Z
        
        [
        η
        ]
        [
        i
        ,
        j
        ,
        
          j
          ′
        
        ]
      
    
    {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }=\mathbb {Z} [\eta ][i,j,j']}
  
.  Then 
  
    
      
        
          
            
              Q
            
          
          
            
              H
              u
              r
            
          
        
      
    
    {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}
  
 is a maximal <a href="/facts/Order_(ring_theory)/4JzIeDKz">order</a> of 
  
    
      
        D
      
    
    {\displaystyle D}
  
 (see <a href="/facts/Hurwitz_quaternion_order/S9ryNPyn">Hurwitz quaternion order</a>), described explicitly by <a href="/facts/Noam_Elkies/t1Uerz9E">Noam Elkies</a> [1].
</p><p>In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in 
  
    
      
        
          Z
        
        [
        η
        ]
      
    
    {\displaystyle \mathbb {Z} [\eta ]}
  
, namely 
</p>

13
        =
        η
        (
        η
        +
        2
        )
        (
        2
        η
        −
        1
        )
        (
        3
        −
        2
        η
        )
        (
        η
        +
        3
        )
        ,
      
    
    {\displaystyle 13=\eta (\eta +2)(2\eta -1)(3-2\eta )(\eta +3),}

<p>where 
  
    
      
        η
        (
        η
        +
        2
        )
      
    
    {\displaystyle \eta (\eta +2)}
  
 is invertible. Also consider the prime ideals generated by the non-invertible factors. The principal congruence subgroup defined by such a prime ideal <i>I</i> is by definition the group
</p>

Q
            
          
          
            
              H
              u
              r
            
          
          
            1
          
        
        (
        I
        )
        =
        {
        x
        ∈
        
          
            
              Q
            
          
          
            
              H
              u
              r
            
          
          
            1
          
        
        :
        x
        ≡
        1
        
          
          (
          mod
          
          I
          
            
              
                Q
              
            
            
              
                H
                u
                r
              
            
          
          )
        
        }
        ,
      
    
    {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }^{1}(I)=\{x\in {\mathcal {Q}}_{\mathrm {Hur} }^{1}:x\equiv 1{\pmod {I{\mathcal {Q}}_{\mathrm {Hur} }}}\},}

<p>namely, the group of elements of <a href="/facts/Reduced_norm/RsBMUkIW">reduced norm</a> 1 in 
  
    
      
        
          
            
              Q
            
          
          
            
              H
              u
              r
            
          
        
      
    
    {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}
  
 equivalent to 1 modulo the ideal 
  
    
      
        I
        
          
            
              Q
            
          
          
            
              H
            
            u
            r
          
        
      
    
    {\displaystyle I{\mathcal {Q}}_{\mathrm {H} ur}}
  
.  The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to P<a href="/facts/SL(2,R)/xjKM9JLJ">SL(2,R)</a>.
</p><p>Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a <a href="/facts/Fuchsian_model/SsXhYMAg">Fuchsian model</a>, the quotient of the <a href="/facts/Hyperbolic_manifold/eUBhz9uW">hyperbolic plane</a> by one of these three Fuchsian groups.
</p>
<h2 id="bound-for-systolic-length-and-the-systolic-ratio">Bound for systolic length and the systolic ratio</h2>
<p>The <a href="/facts/Gauss%E2%80%93Bonnet_theorem/pvTbtH91">Gauss–Bonnet theorem</a> states that 
</p>

χ
        (
        Σ
        )
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            Σ
          
        
        K
        (
        u
        )
        
        d
        A
        ,
      
    
    {\displaystyle \chi (\Sigma )={\frac {1}{2\pi }}\int _{\Sigma }K(u)\,dA,}

<p>where 
  
    
      
        χ
        (
        Σ
        )
      
    
    {\displaystyle \chi (\Sigma )}
  
 is the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of the surface and 
  
    
      
        K
        (
        u
        )
      
    
    {\displaystyle K(u)}
  
 is the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> . In the case 
  
    
      
        g
        =
        14
      
    
    {\displaystyle g=14}
  
 we have
</p>

χ
        (
        Σ
        )
        =
        −
        26
      
    
    {\displaystyle \chi (\Sigma )=-26}
  
 and 
  
    
      
        K
        (
        u
        )
        =
        −
        1
        ,
      
    
    {\displaystyle K(u)=-1,}

<p>thus we obtain that the area of these surfaces is 
</p>

52
        π
      
    
    {\displaystyle 52\pi }
  
.
<p>The lower bound on the <a href="/facts/Systolic_geometry/exKrWG9e">systole</a> as specified in [2], namely 
</p>

4
            3
          
        
        log
        ⁡
        (
        g
        (
        Σ
        )
        )
        ,
      
    
    {\displaystyle {\frac {4}{3}}\log(g(\Sigma )),}

<p>is 3.5187.
</p><p>Some specific details about each of the surfaces are presented in the following tables (the number of systolic loops is taken from [3]). The term Systolic Trace refers to the least reduced trace of an element in the corresponding subgroup 
  
    
      
        
          
            
              Q
            
          
          
            H
            u
            r
          
          
            1
          
        
        (
        I
        )
      
    
    {\displaystyle {\mathcal {Q}}_{Hur}^{1}(I)}
  
.  The systolic ratio is the ratio of the square of the systole to the area.
</p>
<table><tbody><tr><td>Ideal</td><td>                    3        −        2        η        ⊲                  O                      K                                {\displaystyle 3-2\eta \vartriangleleft O_{K}}  </td></tr><tr><td>Systole</td><td>5.9039</td></tr><tr><td>Systolic Trace</td><td>                    −        4                  η                      2                          −        8        η        −        3              {\displaystyle -4\eta ^{2}-8\eta -3}  </td></tr><tr><td>Systolic Ratio</td><td>0.2133</td></tr><tr><td>Number of Systolic Loops</td><td>91</td></tr></tbody></table>
<table><tbody><tr><td>Ideal</td><td>                    η        +        3        ⊲                  O                      K                                {\displaystyle \eta +3\vartriangleleft O_{K}}  </td></tr><tr><td>Systole</td><td>6.3933</td></tr><tr><td>Systolic Trace</td><td>                    5                  η                      2                          +        11        η        +        3              {\displaystyle 5\eta ^{2}+11\eta +3}  </td></tr><tr><td>Systolic Ratio</td><td>0.2502</td></tr><tr><td>Number of Systolic Loops</td><td>78</td></tr></tbody></table>
<table><tbody><tr><td>Ideal</td><td>                    2        η        −        1        ⊲                  O                      K                                {\displaystyle 2\eta -1\vartriangleleft O_{K}}  </td></tr><tr><td>Systole</td><td>6.8879</td></tr><tr><td>Systolic Trace</td><td>                    −        7                  η                      2                          −        14        η        −        3              {\displaystyle -7\eta ^{2}-14\eta -3}  </td></tr><tr><td>Systolic Ratio</td><td>0.2904</td></tr><tr><td>Number of Systolic Loops</td><td>364</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/(2,3,7)_triangle_group/vqpzGzTA">(2,3,7) triangle group</a></li></ul>

<ul><li><a href="/facts/Noam_Elkies/t1Uerz9E">Elkies, N.</a> (1999). <i>The Klein quartic in number theory. The eightfold way</i>. Math. Sci. Res. Inst. Publ. Vol. 35. Cambridge: Cambridge Univ. Press. pp. 51–101.</li>
<li>Katz, M.; Schaps, M.; Vishne, U. (2007). "Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups". <i>J. Differential Geom</i>. 76 (3): 399–422. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.DG/0505007">math.DG/0505007</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2Fjdg%2F1180135693">10.4310/jdg/1180135693</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:18152345">18152345</a>.</li>
<li>Vogeler, R. (2003). <i>On the geometry of Hurwitz surfaces</i> (Thesis). Florida State University.</li></ul>

First Hurwitz triplet open-in-new

First Hurwitz triplet