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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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    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.2010-11-22

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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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    @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.2010-09-18
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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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    .. presumably a nephew of emperor Charlemagne, a legendary hero in French mythology, and the central character of Tasso's epic poem "Orlando furioso"), a fabulous animal which had all the qualities in the world except... existence. Only in 1985, with all the theoretical progress made in more than 10 years, Frey was able to state his conjecture that "Roland's mare" could not be modular. See the historical appendix of Hell.'s book.2017-11-26