Moduli of abelian varieties

<p><a href="/facts/Abelian_varieties/dGSDVruG">Abelian varieties</a> are a natural generalization of <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a> to higher dimensions.  However, unlike the case of elliptic curves, there is no well-behaved stack playing the role of a <a href="/facts/Moduli_space/9sqrdpwR">moduli stack</a> for higher-dimensional abelian varieties.  One can solve this problem by constructing a moduli stack of abelian varieties equipped with extra structure, such as a <a href="/facts/Abelian_variety/dGSDVruG">principal polarisation</a>.  Just as there is a <a href="/facts/Moduli_stack_of_elliptic_curves/n4sE3GSp">moduli stack</a> of elliptic curves over 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
 constructed as a stacky quotient of the <a href="/facts/Upper_half-plane/7fPqrowr">upper-half plane</a> by the action of 
  
    
      
        S
        
          L
          
            2
          
        
        (
        
          Z
        
        )
      
    
    {\displaystyle SL_{2}(\mathbb {Z} )}
  
, there is a moduli space of principally polarised abelian varieties given as a stacky quotient of <a href="/facts/Siegel_upper_half-space/UlvKRxUs">Siegel upper half-space</a> by the <a href="/facts/Symplectic_group/G4CKBc7z">symplectic group</a> 
  
    
      
        
          Sp
          
            2
            g
          
        
        ⁡
        (
        
          Z
        
        )
      
    
    {\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
  
.   By adding even more extra structure, such as a level <i>n</i> structure, one can go further and obtain a <a href="/facts/Moduli_space/9sqrdpwR">fine moduli space</a>.
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Moduli of abelian varieties open-in-new

Moduli of abelian varieties