Semistable abelian variety

<h2 id="semistable-elliptic-curve">Semistable elliptic curve</h2>
A semistable elliptic curve may be described more concretely as an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> that has <a href="/facts/Bad_reduction/MlmGW0Gm">bad reduction</a> only of multiplicative type.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Suppose E is an elliptic curve defined over the <a href="/facts/Rational_number/VJQmyY05">rational number</a> field 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
. It is known that there is a <a href="/facts/Finite_set/za1newlP">finite</a>, <a href="/facts/Non-empty_set/ffEk3eA6">non-empty set</a> S of <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> p for which E has bad reduction <a href="/facts/Modular_arithmetic/0wtRa5dU">modulo</a> p. The latter means that the curve 
 
 
 
 
 E
 
 p
 
 
 
 
 {\displaystyle E_{p}}
 
 obtained by reduction of E to the <a href="/facts/Prime_field/D2VzcQaG">prime field</a> with p elements has a <a href="/facts/Mathematical_singularity/Y5sMQY4I">singular point</a>. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a <a href="/facts/Double_point/mGFKPaas">double point</a>, rather than a <a href="/facts/Cusp_(singularity)/QH3fEGry">cusp</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Deciding whether this condition holds is effectively computable by <a href="/facts/Tate%27s_algorithm/dVePyx1L">Tate's algorithm</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Therefore in a given case it is decidable whether or not the reduction is semistable, namely multiplicative reduction at worst.
The semistable reduction theorem for E may also be made explicit: E acquires semistable reduction over the extension of F generated by the coordinates of the points of order 12.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<ul><li><a href="/facts/Alexandre_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a> (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1. Lecture Notes in Mathematics (in French). Vol. 288. Berlin; New York: <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer-Verlag</a>. viii+523. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0068688">10.1007/BFb0068688</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05987-5. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0354656">0354656</a>.</li>
<li>Husemöller, Dale H. (1987). Elliptic curves. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 111. With an appendix by <a href="/facts/Ruth_Lawrence/onYL6Ru9">Ruth Lawrence</a>. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-96371-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0605.14032">0605.14032</a>.</li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1997). <a href="https://archive.org/details/surveydiophantin00lang_347">Survey of Diophantine geometry</a>. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. p. <a href="https://archive.org/details/surveydiophantin00lang_347/page/n83">70</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-61223-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0869.11051">0869.11051</a>.</li></ul>

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

<ol>
<li id="fn:1">Grothendieck (1972) Théorème 3.6, p. 351 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Husemöller (1987) pp.116-117 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Husemoller (1987) pp.116-117 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Husemöller (1987) pp.266-269 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Tate, John (1975), "Algorithm for determining the type of a singular fiber in an elliptic pencil", in Birch, B.J.; Kuyk, W. (eds.), Modular Functions of One Variable IV, Lecture Notes in Mathematics, vol. 476, Berlin / Heidelberg: Springer, pp. 33–52, doi:10.1007/BFb0097582, ISBN 978-3-540-07392-5, ISSN 1617-9692, MR 0393039, Zbl 1214.14020 <a href="978-3-540-07392-5" target="_blank">978-3-540-07392-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">This is implicit in Husemöller (1987) pp.117-118 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Tate, John (1975), "Algorithm for determining the type of a singular fiber in an elliptic pencil", in Birch, B.J.; Kuyk, W. (eds.), Modular Functions of One Variable IV, Lecture Notes in Mathematics, vol. 476, Berlin / Heidelberg: Springer, pp. 33–52, doi:10.1007/BFb0097582, ISBN 978-3-540-07392-5, ISSN 1617-9692, MR 0393039, Zbl 1214.14020 <a href="978-3-540-07392-5" target="_blank">978-3-540-07392-5</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Semistable abelian variety open-in-new

Semistable abelian variety