Siegel's theorem on integral points

<h2 id="statement">Statement</h2>
<blockquote><p>Siegel's theorem on integral points: For a <a href="/facts/Singular_point_of_an_algebraic_variety/8nvkrkE9">smooth</a> <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> <i>C</i> of <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> <i>g</i> defined over a <a href="/facts/Number_field/e0K0TXSh">number field</a> <i>K</i>, presented in <a href="/facts/Affine_space/tlvI3oX1">affine space</a> in a given coordinate system, there are only finitely many points on <i>C</i> with coordinates in the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> <i>O</i> of <i>K</i>, provided <i>g</i> > 0.</p></blockquote>
<h2 id="history">History</h2>
<p>In 1926, Siegel proved the theorem effectively in the special case 
  
    
      
        g
        =
        1
      
    
    {\displaystyle g=1}
  
, so that he proved this theorem conditionally, provided the <a href="/facts/Mordell%27s_conjecture/GhyRaknq">Mordell's conjecture</a> is true.
</p><p>In 1929, Siegel proved the theorem unconditionally by combining a version of the <a href="/facts/Thue%E2%80%93Siegel%E2%80%93Roth_theorem/ll7ubGmd">Thue–Siegel–Roth theorem</a>, from <a href="/facts/Diophantine_approximation/WGVMuWF4">diophantine approximation</a>, with the <a href="/facts/Mordell%E2%80%93Weil_theorem/3ko0yKew">Mordell–Weil theorem</a> from <a href="/facts/Diophantine_geometry/H2x4wzUE">diophantine geometry</a> (required in Weil's version, to apply to the <a href="/facts/Jacobian_variety/HFedCxIn">Jacobian variety</a> of <i>C</i>).
</p><p>In 2002, <a href="/facts/Umberto_Zannier/dFhTXSdf">Umberto Zannier</a> and <a href="/facts/Pietro_Corvaja/CkNXD9R7">Pietro Corvaja</a> gave a new proof by using a new method based on the <a href="/facts/Subspace_theorem/Y8vcqDzH">subspace theorem</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="effective-versions">Effective versions</h2>
<p>Siegel's result was ineffective for 
  
    
      
        g
        ≥
        2
      
    
    {\displaystyle g\geq 2}
  
 (see <a href="/facts/Effective_results_in_number_theory/uM8p8XGf">effective results in number theory</a>), since <a href="/facts/Axel_Thue/rBoRr6Qd">Thue</a>'s method in diophantine approximation also is ineffective in describing possible very good rational approximations to almost all <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a> of degree 
  
    
      
        d
        ≥
        5
      
    
    {\displaystyle d\geq 5}
  
. Siegel proved it effectively only in the special case 
  
    
      
        g
        =
        1
      
    
    {\displaystyle g=1}
  
 in 1926. Effective results in some cases derive from <a href="/facts/Baker%27s_method/GXcd8RbV">Baker's method</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Diophantine_geometry/H2x4wzUE">Diophantine geometry</a></li></ul>

<ul><li><a href="/facts/Enrico_Bombieri/MU6xU5vr">Bombieri, Enrico</a>; Gubler, Walter (2006). <i>Heights in Diophantine Geometry</i>. New Mathematical Monographs. Vol. 4. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-71229-3. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1130.11034">1130.11034</a>.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1978). <i>Elliptic curves: Diophantine analysis</i>. Grundlehren der mathematischen Wissenschaften. Vol. 231. pp. 128–153. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-08489-4. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0388.10001">0388.10001</a>.</li>
<li><a href="/facts/Carl_Ludwig_Siegel/I91aMunI">Siegel, Carl Ludwig</a> (1929). "Über einige Anwendungen diophantischer Approximationen". <i>Sitzungsberichte der Preussischen Akademie der Wissenschaften</i> (in German).</li></ul>

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

<ol>
<li id="fn:1"><p>Corvaja, P. and Zannier, U. "A subspace theorem approach to integral points on curves", Compte Rendu Acad. Sci., 334, 2002, pp. 267–271 doi:10.1016/S1631-073X(02)02240-9 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Siegel's theorem on integral points open-in-new

Siegel's theorem on integral points