Bogomolov conjecture

<h2 id="statement">Statement</h2>
<p>Let <i>C</i> be an <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> of <a href="/facts/Genus_(curve)/o1Yhf8ty">genus</a> <i>g</i> at least two defined over a <a href="/facts/Number_field/e0K0TXSh">number field</a> <i>K</i>, let 
  
    
      
        
          
            K
            ¯
          
        
      
    
    {\displaystyle {\overline {K}}}
  
 denote the <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of <i>K</i>, fix an embedding of <i>C</i> into its <a href="/facts/Jacobian_variety/HFedCxIn">Jacobian variety</a> <i>J</i>, and let 
  
    
      
        
          
            
              h
              ^
            
          
        
      
    
    {\displaystyle {\hat {h}}}
  
 denote the <a href="/facts/N%C3%A9ron-Tate_height/IpAkTcqU">Néron-Tate height</a> on <i>J</i> associated to an <a href="/facts/Ample_divisor/XRQKvXGD">ample symmetric divisor</a>. Then there exists an 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
  
 such that the set
</p>

{
        P
        ∈
        C
        (
        
          
            K
            ¯
          
        
        )
        :
        
          
            
              h
              ^
            
          
        
        (
        P
        )
        <
        ϵ
        }
      
    
    {\displaystyle \{P\in C({\overline {K}}):{\hat {h}}(P)<\epsilon \}}
  
    is finite.
<p>Since 
  
    
      
        
          
            
              h
              ^
            
          
        
        (
        P
        )
        =
        0
      
    
    {\displaystyle {\hat {h}}(P)=0}
  
 if and only if <i>P</i> is a <a href="/facts/Torsion_point/uNRnwtLz">torsion point</a>, the Bogomolov conjecture generalises the <a href="/facts/Manin-Mumford_conjecture/7g7zQ4CN">Manin-Mumford conjecture</a>.
</p>
<h2 id="proof">Proof</h2>
<p>The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang using <a href="/facts/Arakelov_theory/JwpLlcMw">Arakelov theory</a> in 1998.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="generalization">Generalization</h2>
<p>In 1998, Zhang proved the following generalization:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Let <i>A</i> be an <a href="/facts/Abelian_variety/dGSDVruG">abelian variety</a> defined over <i>K</i>, and let 
  
    
      
        
          
            
              h
              ^
            
          
        
      
    
    {\displaystyle {\hat {h}}}
  
 be the Néron-Tate height on <i>A</i> associated to an ample symmetric divisor. A subvariety 
  
    
      
        X
        ⊂
        A
      
    
    {\displaystyle X\subset A}
  
 is called a <i>torsion subvariety</i> if it is the translate of an abelian subvariety of <i>A</i> by a torsion point. If <i>X</i> is not a torsion subvariety, then there is an 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
  
 such that the set
</p>

{
        P
        ∈
        X
        (
        
          
            K
            ¯
          
        
        )
        :
        
          
            
              h
              ^
            
          
        
        (
        P
        )
        <
        ϵ
        }
      
    
    {\displaystyle \{P\in X({\overline {K}}):{\hat {h}}(P)<\epsilon \}}
  
    is not <a href="/facts/Zariski_dense/uolkKBpM">Zariski dense</a> in <i>X</i>.

<h3>Other sources</h3>
<ul><li>Chambert-Loir, Antoine (2013). "Diophantine geometry and analytic spaces". In Amini, Omid; Baker, Matthew; Faber, Xander (eds.). <i>Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011</i>. Contemporary Mathematics. Vol. 605. Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. pp. 161–179. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4704-1021-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1281.14002">1281.14002</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="https://web.archive.org/web/20140508042701/http://modular.math.washington.edu/home/wstein/www/home/bober/swc/www/aws/1999/99Tzermias.pdf">The Manin-Mumford conjecture: a brief survey, by Pavlos Tzermias</a></li></ul>

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

<ol>
<li id="fn:1"><p>Ullmo, Emmanuel (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics, 147 (1): 167–179, arXiv:alg-geom/9606017, doi:10.2307/120987, JSTOR 120987, Zbl 0934.14013. <a href="/wiki/Annals_of_Mathematics" target="_blank">/wiki/Annals_of_Mathematics</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986 <a href="/wiki/Annals_of_Mathematics" target="_blank">/wiki/Annals_of_Mathematics</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986 <a href="/wiki/Annals_of_Mathematics" target="_blank">/wiki/Annals_of_Mathematics</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Bogomolov conjecture open-in-new

Bogomolov conjecture