Moduli stack of elliptic curves

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the moduli stack of elliptic curves, denoted as 
  
    
      
        
          
            
              M
            
          
          
            1
            ,
            1
          
        
      
    
    {\displaystyle {\mathcal {M}}_{1,1}}
  
 or 
  
    
      
        
          
            
              M
            
          
          
            
              e
              l
              l
            
          
        
      
    
    {\displaystyle {\mathcal {M}}_{\mathrm {ell} }}
  
, is an <a href="/facts/Algebraic_stack/flrtwvwU">algebraic stack</a> over 
  
    
      
        
          Spec
        
        (
        
          Z
        
        )
      
    
    {\displaystyle {\text{Spec}}(\mathbb {Z} )}
  
 classifying <a href="/facts/Elliptic_curves/60jvaHSA">elliptic curves</a>. Note that it is a special case of the <a href="/facts/Moduli_stack_of_algebraic_curves/u4mAMEfJ">moduli stack of algebraic curves</a> 
  
    
      
        
          
            
              M
            
          
          
            g
            ,
            n
          
        
      
    
    {\displaystyle {\mathcal {M}}_{g,n}}
  
. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme 
  
    
      
        S
      
    
    {\displaystyle S}
  
 to it correspond to elliptic curves over 
  
    
      
        S
      
    
    {\displaystyle S}
  
. The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in 
  
    
      
        
          
            
              M
            
          
          
            1
            ,
            1
          
        
      
    
    {\displaystyle {\mathcal {M}}_{1,1}}
  
.
</p>

Moduli stack of elliptic curves open-in-new

Moduli stack of elliptic curves