A question on FLT and Taniyama Shimura

Solution 1:

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.)

Solution 2:

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 $2$.