Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?

2010-08-31T03:18:40Z

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.

One way to prove this, which Cremona and Lingham use here 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.

Here's a proof with overkill:

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.

The corresponding spaces $S_2(\Gamma_0(N))$ of cuspforms for each of our finite list of $N$ is finite dimensional.

By the modularity theorem, there is hence finite number isogeny classes of elliptic curves with everywhere good reduction outside $S$.

By Mazur's Modular Curves and the Eisenstein Ideal there are only a finite number of isomorphism classes of elliptic curves in a given isogeny class.

Question 1: Does any of this machinery rely on Siegel's theorem?

Question 2: 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?

By "cheaply extended" I mean without the use of techniques with the diophantine flavor of Baker's theory of linear forms in logarithms.

Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper? - MathOverflow

Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?