Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:21:38Z http://mathoverflow.net/feeds/question/37212 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37212/can-you-get-siegels-theorem-for-free-from-modularity-and-mazurs-eisenstein-id Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper? Jamie Weigandt 2010-08-31T03:18:40Z 2010-08-31T05:50:20Z <p>There is a well-known theorem of Shafarevich that given a finite set \$S\$ of primes the number of isomorphism classes of elliptic curves over \$\Bbb Q\$ with everywhere good reduction outside \$S\$ is finite.</p> <p>One way to prove this, which Cremona and Lingham use <a href="http://www.warwick.ac.uk/~masgaj/papers/egros.pdf" rel="nofollow">here</a> to compute all such curves, is to use Siegel's theorem that an elliptic curve over \$Q\$ has only a finite number of \$S\$-integral points.</p> <p>Here's a proof with overkill:</p> <p>Given \$S\$ there are a finite number of possible conductors \$N\$ for elliptic curves with everywhere good reduction outside \$S\$. They must all be divisors of \$2^8 3^5 d^2\$ where \$d\$ is the product of those primes in \$S\$ different from 2 and 3.</p> <p>The corresponding spaces \$S_2(\Gamma_0(N))\$ of cuspforms for each of our finite list of \$N\$ is finite dimensional.</p> <p>By the modularity theorem, there is hence finite number isogeny classes of elliptic curves with everywhere good reduction outside \$S\$.</p> <p>By Mazur's <em>Modular Curves and the Eisenstein Ideal</em> there are only a finite number of isomorphism classes of elliptic curves in a given isogeny class.</p> <blockquote> <p><strong>Question 1:</strong> Does any of this machinery rely on Siegel's theorem? </p> <p><strong>Question 2:</strong> If the answer to question 1 is no, can this proof of Shafarevich's theorem be "cheaply extended" to deduce Siegel's Theorem from these seemingly unrelated powerful results?</p> </blockquote> <p>By "cheaply extended" I mean without the use of techniques with the diophantine flavor of Baker's theory of linear forms in logarithms.</p>