Siegel modular form

<h2 id="definition">Definition</h2>
<h3>Preliminaries</h3>
<p>Let 
  
    
      
        g
        ,
        N
        ∈
        
          N
        
      
    
    {\displaystyle g,N\in \mathbb {N} }
  
 and define
</p>

H
            
          
          
            g
          
        
        =
        
          {
          
            τ
            ∈
            
              M
              
                g
                ×
                g
              
            
            (
            
              C
            
            )
             
            
              
                |
              
            
             
            
              τ
              
                
                  T
                
              
            
            =
            τ
            ,
            
              
                Im
              
            
            (
            τ
            )
            
               positive definite
            
          
          }
        
        ,
      
    
    {\displaystyle {\mathcal {H}}_{g}=\left\{\tau \in M_{g\times g}(\mathbb {C} )\ {\big |}\ \tau ^{\mathrm {T} }=\tau ,{\textrm {Im}}(\tau ){\text{  positive definite}}\right\},}

<p>the <a href="/facts/Siegel_upper_half-space/UlvKRxUs">Siegel upper half-space</a>. Define the <a href="/facts/Symplectic_group/G4CKBc7z">symplectic group</a> of level 
  
    
      
        N
      
    
    {\displaystyle N}
  
, denoted by 
  
    
      
        
          Γ
          
            g
          
        
        (
        N
        )
        ,
      
    
    {\displaystyle \Gamma _{g}(N),}
  
 as 
</p>

Γ
          
            g
          
        
        (
        N
        )
        =
        
          {
          
            γ
            ∈
            G
            
              L
              
                2
                g
              
            
            (
            
              Z
            
            )
             
            
              
                |
              
            
             
            
              γ
              
                
                  T
                
              
            
            
              
                (
                
                  
                    
                      0
                    
                    
                      
                        I
                        
                          g
                        
                      
                    
                  
                  
                    
                      −
                      
                        I
                        
                          g
                        
                      
                    
                    
                      0
                    
                  
                
                )
              
            
            γ
            =
            
              
                (
                
                  
                    
                      0
                    
                    
                      
                        I
                        
                          g
                        
                      
                    
                  
                  
                    
                      −
                      
                        I
                        
                          g
                        
                      
                    
                    
                      0
                    
                  
                
                )
              
            
            ,
             
            γ
            ≡
            
              I
              
                2
                g
              
            
            
            mod
            
            
            N
          
          }
        
        ,
      
    
    {\displaystyle \Gamma _{g}(N)=\left\{\gamma \in GL_{2g}(\mathbb {Z} )\ {\big |}\ \gamma ^{\mathrm {T} }{\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}}\gamma ={\begin{pmatrix}0&I_{g}\\-I_{g}&0\end{pmatrix}},\ \gamma \equiv I_{2g}\mod N\right\},}

<p>where 
  
    
      
        
          I
          
            g
          
        
      
    
    {\displaystyle I_{g}}
  
 is the 
  
    
      
        g
        ×
        g
      
    
    {\displaystyle g\times g}
  
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. Finally, let 
</p>

ρ
        :
        
          
            
              GL
            
          
          
            g
          
        
        (
        
          C
        
        )
        →
        
          
            GL
          
        
        (
        V
        )
      
    
    {\displaystyle \rho :{\textrm {GL}}_{g}(\mathbb {C} )\rightarrow {\textrm {GL}}(V)}

<p>be a <a href="/facts/Rational_representation/WNfkDD1G">rational representation</a>, where 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is a finite-dimensional complex <a href="/facts/Vector_space/M1pxMLx2">vector space</a>.
</p>
<h3>Siegel modular form</h3>
<p>Given 
</p>

γ
        =
        
          
            (
            
              
                
                  A
                
                
                  B
                
              
              
                
                  C
                
                
                  D
                
              
            
            )
          
        
      
    
    {\displaystyle \gamma ={\begin{pmatrix}A&B\\C&D\end{pmatrix}}}

γ
        ∈
        
          Γ
          
            g
          
        
        (
        N
        )
        ,
      
    
    {\displaystyle \gamma \in \Gamma _{g}(N),}

<p>define the notation 
</p>

(
        f
        
          
            |
          
        
        γ
        )
        (
        τ
        )
        =
        (
        ρ
        (
        C
        τ
        +
        D
        )
        
          )
          
            −
            1
          
        
        f
        (
        γ
        τ
        )
        .
      
    
    {\displaystyle (f{\big |}\gamma )(\tau )=(\rho (C\tau +D))^{-1}f(\gamma \tau ).}

<p>Then a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a>
</p>

f
        :
        
          
            
              H
            
          
          
            g
          
        
        →
        V
      
    
    {\displaystyle f:{\mathcal {H}}_{g}\rightarrow V}

<p>is a <i>Siegel modular form</i> of degree 
  
    
      
        g
      
    
    {\displaystyle g}
  
 (sometimes called the genus), weight 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
, and level 
  
    
      
        N
      
    
    {\displaystyle N}
  
 if 
</p>

(
        f
        
          
            |
          
        
        γ
        )
        =
        f
      
    
    {\displaystyle (f{\big |}\gamma )=f}

<p>for all 
  
    
      
        γ
        ∈
        
          Γ
          
            g
          
        
        (
        N
        )
      
    
    {\displaystyle \gamma \in \Gamma _{g}(N)}
  
.
In the case that 
  
    
      
        g
        =
        1
      
    
    {\displaystyle g=1}
  
, we further require that 
  
    
      
        f
      
    
    {\displaystyle f}
  
 be holomorphic 'at infinity'. This assumption is not necessary for 
  
    
      
        g
        >
        1
      
    
    {\displaystyle g>1}
  
 due to the Koecher principle, explained below. Denote the space of weight 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
, degree 
  
    
      
        g
      
    
    {\displaystyle g}
  
, and level 
  
    
      
        N
      
    
    {\displaystyle N}
  
 Siegel modular forms by 
</p>

M
          
            ρ
          
        
        (
        
          Γ
          
            g
          
        
        (
        N
        )
        )
        .
      
    
    {\displaystyle M_{\rho }(\Gamma _{g}(N)).}

<h2 id="examples">Examples</h2>
<p>Some methods for constructing Siegel modular forms include:
</p>
<ul><li><a href="/facts/Eisenstein_series/23vgJLFl">Eisenstein series</a></li>
<li><a href="/facts/Theta_function_of_a_lattice/7k3KN9nJ">Theta functions of lattices</a> and <a href="/facts/Siegel_theta_series/o8tlDXJa">Siegel theta series</a></li>
<li><a href="/facts/Saito%E2%80%93Kurokawa_lift/0jvntrKp">Saito–Kurokawa lift</a> for degree 2</li>
<li><a href="/facts/Ikeda_lift/ceHcv7lI">Ikeda lift</a></li>
<li><a href="/facts/Miyawaki_lift/mSX2Vqfs">Miyawaki lift</a></li>
<li>Products of Siegel modular forms.</li></ul>
<h3>Level 1, small degree</h3>
<p>For degree 1, the level 1 Siegel modular forms are the same as level 1 modular forms. The ring of such forms is a polynomial ring C[<i>E</i>4,<i>E</i>6] in the (degree 1) Eisenstein series <i>E</i>4 and <i>E</i>6.
</p><p>For degree 2, (Igusa 1962, 1967) showed that the ring of level 1 Siegel modular forms is generated by the (degree 2) Eisenstein series <i>E</i>4 and <i>E</i>6 and 3 more forms of weights 10, 12, and 35. The ideal of relations between them is generated by the square of the weight 35 form minus a certain polynomial in the others.
</p><p>For degree 3, Tsuyumine (1986) described the ring of level 1 Siegel modular forms, giving a set of 34 generators.
</p><p>For degree 4, the level 1 Siegel modular forms of small weights have been found. There are no cusp forms of weights 2, 4, or 6. The space of cusp forms of weight 8 is 1-dimensional, spanned by the <a href="/facts/Schottky_form/4hobTXoo">Schottky form</a>. The space of cusp forms of weight 10 has dimension 1, the space of cusp forms of weight 12 has dimension 2, the space of cusp forms of weight 14 has dimension 3, and the space of cusp forms of weight 16 has dimension 7 (Poor & Yuen 2007) harv error: no target: CITEREFPoorYuen2007 (help).
</p><p>For degree 5, the space of cusp forms has dimension 0 for weight 10, dimension 2 for weight 12. The space of forms of weight 12 has dimension 5.
</p><p>For degree 6, there are no cusp forms of weights 0, 2, 4, 6, 8. The space of Siegel modular forms of weight 2 has dimension 0, and those of weights 4 or 6 both have dimension 1.
</p>
<h3>Level 1, small weight</h3>
<p>For small weights and level 1, Duke & Imamoḡlu (1998) give the following results (for any positive degree):
</p>
<ul><li>Weight 0: The space of forms is 1-dimensional, spanned by 1.</li>
<li>Weight 1: The only Siegel modular form is 0.</li>
<li>Weight 2: The only Siegel modular form is 0.</li>
<li>Weight 3: The only Siegel modular form is 0.</li>
<li>Weight 4: For any degree, the space of forms of weight 4 is 1-dimensional, spanned by the theta function of the E8 lattice (of appropriate degree). The only cusp form is 0.</li>
<li>Weight 5: The only Siegel modular form is 0.</li>
<li>Weight 6: The space of forms of weight 6 has dimension 1 if the degree is at most 8, and dimension 0 if the degree is at least 9. The only cusp form is 0.</li>
<li>Weight 7: The space of cusp forms vanishes if the degree is 4 or 7.</li>
<li>Weight 8:In genus 4, the space of cusp forms is 1-dimensional, spanned by the <a href="/facts/Schottky_form/4hobTXoo">Schottky form</a> and the space of forms is 2-dimensional. There are no cusp forms if the genus is 8.</li>
<li>There are no cusp forms if the genus is greater than twice the weight.</li></ul>
<h3>Table of dimensions of spaces of level 1 Siegel modular forms</h3>
<p>The following table combines the results above with information from Poor & Yuen (2006) harvtxt error: no target: CITEREFPoorYuen2006 (help) and Chenevier & Lannes (2014) and Taïbi (2014).
</p>
Dimensions of spaces of level 1 Siegel cusp forms:  Siegel modular forms<table><tbody><tr><th>Weight</th><th>degree 0</th><th>degree 1</th><th>degree 2</th><th>degree 3</th><th>degree 4</th><th>degree 5</th><th>degree 6</th><th>degree 7</th><th>degree 8</th><th>degree 9</th><th>degree 10</th><th>degree 11</th><th>degree 12</th></tr><tr><td>0</td><td>1: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td></tr><tr><td>2</td><td>1: 1</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td></tr><tr><td>4</td><td>1: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td></tr><tr><td>6</td><td>1: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 1</td><td>0: 0</td><td>0: 0</td><td>0: 0</td><td>0: 0</td></tr><tr><td>8</td><td>1: 1</td><td>0: 1</td><td>0: 1</td><td>0:1</td><td>1: 2</td><td>0: 2</td><td>0: 2</td><td>0: 2</td><td>0: 2</td><td>0:</td><td>0:</td><td>0:</td><td>0:</td></tr><tr><td>10</td><td>1: 1</td><td>0: 1</td><td>1: 2</td><td>0: 2</td><td>1: 3</td><td>0: 3</td><td>1: 4</td><td>0: 4</td><td>1:</td><td>0:</td><td>0:</td><td>0:</td><td>0:</td></tr><tr><td>12</td><td>1: 1</td><td>1: 2</td><td>1: 3</td><td>1:  4</td><td>2: 6</td><td>2: 8</td><td>3: 11</td><td>3: 14</td><td>4: 18</td><td>2:20</td><td>2: 22</td><td>1: 23</td><td>1: 24</td></tr><tr><td>14</td><td>1: 1</td><td>0: 1</td><td>1: 2</td><td>1: 3</td><td>3:6</td><td>3: 9</td><td>9: 18</td><td>9: 27</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>16</td><td>1: 1</td><td>1: 2</td><td>2: 4</td><td>3: 7</td><td>7: 14</td><td>13:27</td><td>33:60</td><td>83:143</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>18</td><td>1: 1</td><td>1: 2</td><td>2: 4</td><td>4:8</td><td>12:20</td><td>28: 48</td><td>117: 163</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>20</td><td>1: 1</td><td>1: 2</td><td>3: 5</td><td>6: 11</td><td>22: 33</td><td>76: 109</td><td>486:595</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>22</td><td>1: 1</td><td>1: 2</td><td>4: 6</td><td>9:15</td><td>38:53</td><td>186:239</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>24</td><td>1: 1</td><td>2: 3</td><td>5: 8</td><td>14: 22</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>26</td><td>1: 1</td><td>1: 2</td><td>5: 7</td><td>17: 24</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>28</td><td>1: 1</td><td>2: 3</td><td>7: 10</td><td>27: 37</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>30</td><td>1: 1</td><td>2: 3</td><td>8: 11</td><td>34: 45</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<h2 id="koecher-principle">Koecher principle</h2>
<p>The theorem known as the <i>Koecher principle</i> states that if 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a Siegel modular form of weight 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
,  level 1, and degree 
  
    
      
        g
        >
        1
      
    
    {\displaystyle g>1}
  
, then 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is bounded on subsets of 
  
    
      
        
          
            
              H
            
          
          
            g
          
        
      
    
    {\displaystyle {\mathcal {H}}_{g}}
  
 of the form 
</p>

{
          
            τ
            ∈
            
              
                
                  H
                
              
              
                g
              
            
             
            
              |
            
            
              
                Im
              
            
            (
            τ
            )
            >
            ϵ
            
              I
              
                g
              
            
          
          }
        
        ,
      
    
    {\displaystyle \left\{\tau \in {\mathcal {H}}_{g}\ |{\textrm {Im}}(\tau )>\epsilon I_{g}\right\},}

<p>where 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
  
. Corollary to this theorem is the fact that Siegel modular forms of degree 
  
    
      
        g
        >
        1
      
    
    {\displaystyle g>1}
  
 have <a href="/facts/Fourier_expansion/s3fsxPAT">Fourier expansions</a> and are thus holomorphic at infinity.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="applications-to-physics">Applications to physics</h2>
<p>In the D1D5P system of <a href="/facts/Extremal_black_hole/F5SYaqc3">supersymmetric black holes</a> in string theory, the function that naturally captures the microstates of black hole entropy is a Siegel modular form.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In general, Siegel modular forms have been described as having the potential to describe black holes or other gravitational systems.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Siegel modular forms also have uses as generating functions for families of CFT2 with increasing central charge in <a href="/facts/Conformal_field_theory/McVlk4hR">conformal field theory</a>, particularly the hypothetical <a href="/facts/AdS/CFT_correspondence/8Ide9C9m">AdS/CFT correspondence</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

<ul><li>Chenevier, Gaëtan; Lannes, Jean (2014), <i>Formes automorphes et voisins de Kneser des réseaux de Niemeier</i>, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1409.7616">1409.7616</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2014arXiv1409.7616C">2014arXiv1409.7616C</a></li>
<li>Duke, W.; Imamoḡlu, Ö. (1998), "Siegel modular forms of small weight", <i>Math. Ann.</i>, 310 (1): 73–82, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs002080050137">10.1007/s002080050137</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1600030">1600030</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122219495">122219495</a></li>
<li>Freitag, E. (1983), <i>Siegelsche Modulfunktionen</i>, Grundlehren der Mathematischen Wissenschaften, vol. 254. Springer-Verlag, Berlin, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-68649-8">10.1007/978-3-642-68649-8</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-11661-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0871067">0871067</a>{{citation}}:  CS1 maint: location missing publisher (link)</li>
<li>van der Geer, Gerard (2008), "Siegel Modular Forms and Their Applications", <i>The 1-2-3 of modular forms, 181–245</i>, Universitext, Berlin: Springer, pp. 181–245, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0605346">math/0605346</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-74119-0_3">10.1007/978-3-540-74119-0_3</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-74117-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2409679">2409679</a></li>
<li>Igusa, Jun-ichi (1962), "On Siegel modular forms of genus two", <i>Amer. J. Math.</i>, 84 (1): 175–200, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372812">10.2307/2372812</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372812">2372812</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0141643">0141643</a></li>
<li>Klingen, Helmut (2003), <i>Introductory Lectures on Siegel Modular Forms</i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-35052-5</li>
<li>Siegel, Carl Ludwig (1939), "Einführung in die Theorie der Modulfunktionen n-ten Grades", <i>Math. Ann.</i>, 116: 617–657, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01597381">10.1007/bf01597381</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0001251">0001251</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:124337559">124337559</a></li>
<li>Taïbi, Olivier (2014), <i>Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula</i>, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1406.4247">1406.4247</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2014arXiv1406.4247T">2014arXiv1406.4247T</a></li>
<li>Tsuyumine, Shigeaki (1986), "On Siegel modular forms of degree three", <i>Amer. J. Math.</i>, 108 (4): 755–862, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2374517">10.2307/2374517</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2374517">2374517</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0853217">0853217</a></li></ul>

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

<ol>
<li id="fn:1"><p>This was proved by Max Koecher, Zur Theorie der Modulformen n-ten Grades I, Mathematische. Zeitschrift 59 (1954), 455–466. A corresponding principle for Hilbert modular forms was apparently known earlier, after Fritz Gotzky, Uber eine zahlentheoretische Anwendung von Modulfunktionen zweier Veranderlicher, Math. Ann. 100 (1928), pp. 411-37 <a href="/wiki/Max_Koecher" target="_blank">/wiki/Max_Koecher</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Belin, Alexandre; Castro, Alejandra; Gomes, João; Keller, Christoph A. (11 April 2017). "Siegel modular forms and black hole entropy". Journal of High Energy Physics. 2017 (4): 57. arXiv:1611.04588. Bibcode:2017JHEP...04..057B. doi:10.1007/JHEP04(2017)057. S2CID 256037311. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Belin, Alexandre; Castro, Alejandra; Gomes, João; Keller, Christoph A. (11 April 2017). "Siegel modular forms and black hole entropy". Journal of High Energy Physics. 2017 (4): 57. arXiv:1611.04588. Bibcode:2017JHEP...04..057B. doi:10.1007/JHEP04(2017)057. S2CID 256037311. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Belin, Alexandre; Castro, Alejandra; Gomes, João; Keller, Christoph A. (7 November 2018). "Siegel paramodular forms and sparseness in AdS3/CFT2". Journal of High Energy Physics. 2018 (11): 37. arXiv:1805.09336. Bibcode:2018JHEP...11..037B. doi:10.1007/JHEP11(2018)037. S2CID 256040660. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Siegel modular form open-in-new

Siegel modular form