Section conjecture

<p>In <a href="/facts/Anabelian_geometry/zqvz8gqf">anabelian geometry</a>, a branch of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the section conjecture gives a conjectural description of the splittings of the <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> 
  
    
      
        
          π
          
            1
          
        
        (
        X
        )
        →
        Gal
        ⁡
        (
        k
        )
      
    
    {\displaystyle \pi _{1}(X)\to \operatorname {Gal} (k)}
  
, where 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a complete <a href="/facts/Smooth_curve/xBQ0pTBW">smooth curve</a> of genus at least 2 over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        k
      
    
    {\displaystyle k}
  
 that is finitely generated over 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
, in terms of decomposition groups of rational points of 
  
    
      
        X
      
    
    {\displaystyle X}
  
. The conjecture was introduced by <a href="/facts/Grothendieck/jLQGPY7M">Alexander Grothendieck</a> (1997) in a 1983 letter to <a href="/facts/Gerd_Faltings/9cC0L9WP">Gerd Faltings</a>.
</p>

<ul><li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexander</a> (1997), "Brief an G. Faltings", in <a href="/facts/Leila_Schneps/A80WwAkW">Schneps, Leila</a>; Lochak, Pierre (eds.), <i>Geometric Galois actions, 1</i>, London Math. Soc. Lecture Note Ser., vol. 242, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, pp. 49–58, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-59642-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1483108">1483108</a></li></ul>

Section conjecture open-in-new

Section conjecture