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Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain things(infact many) which i couldn't follow and i would like it to be explained here.

The first is:

The Taniyama-Shimura conjecture. In the video it's said that that an elliptic curve is a modular form in disguise. I would want someone to explain this statement. I have seen the definition of a Modular form in Wikipedia, but i can't correlate this with an elliptic curve. The definition of an elliptic curve is simply a cubic equation of the form $y^{2}=x^{3}+ax+b$. How can it be a modular form?

Next, there was a mention of this Mathematician named "Gerhard Frey" who seems to considered this question of what could happen if there was a solution to the equation, $x^{n}+y^{n}=z^{n}$, and by considering this he constructed a curve which is not modular, contradicting Taniyama-Shimura. If he had constructed such a curve then what was the need for Prof. Ribet to actually prove the Epsilon conjecture.

Lastly, here i would like to know this answer: How many of you agree with Wiles, that possibly Fermat could have fooled himself by saying that he had a proof of this result? I certainly disagree with his statement. Well, the reason, is just instinct!

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To be frank, Chandru, this is one of your better questions. :) –  J. M. Sep 18 '10 at 13:24
@Chandru1: Frey did not prove that the Frey curve is not modular. That is what Ribet did. –  Qiaochu Yuan Sep 18 '10 at 19:31
I remember someone guessing that Fermat had proved his theorem for $n = 3$ and he supposed his proof would generalize, while it turns out that it doesn't. –  Gunnar Magnusson Nov 22 '10 at 7:43
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2 Answers

up vote 6 down vote accepted

The elliptic curve is not a modular form. The idea that there is a modular form $f$ associated to $E$. It satisfies $$f(z)=\sum_{n=1}^\infty c_n q^n$$ where $q=\exp(2\pi i z)$, $c_1=1$ and (with finitely many exceptions) for prime $p$, the equation $y^2=x^3+ax=b$ has $p-c_p$ solutions $(x,y)$ considered modulo $p$. It also satisfies various other conditions that I won't spell out (that it's a "newform" for a modular group $\Gamma_0(N)$ etc.)

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Robin, please correct my impression of the whole setup if I'm wrong: what Frey did amounted to constructing a "weird" elliptic curve that, if it did not have an associated modular form, would imply the falsity of TSC and thus FLT. Meaning, what Ribet essentially did to settle the ε conjecture was to show that the Frey curve does have an associated modular form. –  J. M. Sep 18 '10 at 13:22
@J. M.: Ribet proved (a statement that implies) that the Frey curve is not modular. That is the contradiction that allows the Taniyama-Shimura conjecture to imply FLT. –  Qiaochu Yuan Sep 18 '10 at 19:30
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The starting point of the link between elliptic curves and modular forms is the following.

From a topological point of view, elliptic curves are just (2-dimensional) tori, i.e. products $S^1\times S^1$, where $S^1$ is a circle.

A torus has always an invariant never vanishing tangent field. Dually, one can find a non-vanishing invariant differential form $\omega$ on every elliptic curve $E$. It is a useful exercise to write $\omega$ in terms of the coordinates $x$ and $y$ when $E$ is given as a Weierstrass cubic.

If you have a modular parametrization $\pi:X_0(N)\rightarrow E$ you can pull-back the form $\omega$ to $X_0(N)$ and in terms of the coordinate $z$ in the complex upper halfplane $\pi^*(\omega)=f(z)dz$ for some holomorphic function $f(z)$. It is basically immediate that $f(z)$ is a modular form of weight $2$.

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btw, my opinion is that Fermat didn't have a valid proof of his statement. –  Andrea Mori Sep 18 '10 at 13:05
No, that's Weierstrass parametrization. The Weierstrass $\wp$ function and its derivative are meromorphic functions on the complex plane wich are periodic with respect to a lattice $\Lambda$. Thus, they realize an embedding of ${\Bbb C}/\Lambda$ into the projective plane whose image turns out to be a cubic. –  Andrea Mori Sep 18 '10 at 13:27
Weierstrass parametrizations exist for all elliptic curves (where "elliptic curve"="algebraic curve of genus 1") whereas modular parametrizations are more an arithmetic phenomenon (STW conjecture dealt only with elliptic curves defined over the rationals) –  Andrea Mori Sep 18 '10 at 13:30
@J. M.: the Weierstrass function is really a function of two things: a parameter tau which lives on the upper half plane, and another parameter which lives in the elliptic curve C/(1, tau). Generally we speak of the Weierstrass function as if tau were constant (making it an elliptic function), but it is also possible to speak of the Weierstrass function as a Laurent series in its second parameter while varying tau (making its coefficients modular forms). The Weierstrass function in its totality is therefore something more complicated called a Jacobi form. –  Qiaochu Yuan Sep 18 '10 at 19:47
@Chandru1: compared to anything else you're going to read about elliptic curves, yes. –  Qiaochu Yuan Sep 21 '10 at 13:13
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