most recent 30 from http://mathoverflow.net 2010-03-20T01:27:52Z http://mathoverflow.net/feeds/question/3971 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3971/tate-uniformization-of-nonsplit-semistable-elliptic-curves Jay Pottharst 2009-11-03T17:51:30Z 2009-11-03T18:32:05Z

<p>Let <code>E/Q_p</code> be an elliptic curve having split multiplicative reduction. Then the Tate uniformization gives a surjective homomorphism of <code>p</code>-adic analytic groups <code>G_m \to E</code>, with infinite cyclic kernel. Is there an analogue of this fact for <code>E</code> having nonsplit multiplicative reduction, perhaps replacing Gm with a nonsplit torus? E.g., can one uniformize <code>E</code> over the quadratic extension where the reduction splits, and then somehow descend?</p> <p>(My intuition was as follows. Take <code>E/Q_p</code> with nonsplit multiplicative reduction, and let <code>K/Q_p</code> be quadratic so that <code>E</code> becomes split semistable over <code>K</code>, and let <code>E'</code> be the <code>K</code>-twist of <code>E</code> (which has split multiplicative reduction). Then one has a short exact sequence</p> <p><code>0 \to Z \to G_m \to E' \to 0</code></p> <p>(where <code>Z</code> is the constant analytic group of integers). Extending scalars to <code>K</code> then applying Weil restriction of scalars, we get</p> <p><code>0 \to X \to T \to A \to 0</code>,</p> <p>where <code>X</code> is an etale-locally-constant analytic group, <code>T</code> is a torus, and <code>A</code> is an abelian variety, each of rank <code>2</code> in the appropriate sense. The latter short exact sequence contains the former short exact sequence as a sub (direct factor?); the quotient sequence should be something like</p> <p><code>0 \to Z' \to Gm' \to E \to 0</code>,</p> <p>where ' still denotes twisting by <code>K/Q_p</code>. Since <code>Z'</code> has trivial <code>Q_p</code>-points, then, one should have something like <code>G_m'(Q_p) = E(Q_p)</code>, modulo any descent used in forming the quotient.</p> <p>Does this sound sensical?</p> <p>If anyone has access to Google Wave and wants to discuss, I've set up a wave here: <a href="https://wave.google.com/wave/#restored:wave:googlewave.com!w%252BQCn6fZTuZ" rel="nofollow">https://wave.google.com/wave/#restored:wave:googlewave.com!w%252BQCn6fZTuZ</a></p>

http://mathoverflow.net/questions/3971/tate-uniformization-of-nonsplit-semistable-elliptic-curves/3984#3984 Scott Carnahan 2009-11-03T18:29:06Z 2009-11-03T18:29:06Z

<p>Most of it makes sense. Elliptic curves with non-split reduction can be analytically uniformized by the norm torus. There is a "nice" picture of this using the Berkovich spectrum for a non-split torus. I have my doubts about the statement concerning rational points - you should have a Galois cohomology exact sequence.</p>

http://mathoverflow.net/questions/3971/tate-uniformization-of-nonsplit-semistable-elliptic-curves/3985#3985 moonface 2009-11-03T18:32:05Z 2009-11-03T18:32:05Z

<p>A form of this is contained in Silverman, second book, Chapter V, Corollary 5.4. I guess that the image of Gm' in E (at the level of Q_p-points) may have index 2. </p>