Can one explain to me this Theorem

Question

Can one explain to me Theorem 2. (page 3) in this link:

http://www.math.leidenuniv.nl/~evertse/siksek-modular.pdf

I am but confused about the nature the bijection defined in that result.

Answer 1

The result in question is the celebrated Elliptic Modularity Theorem, due to Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor, and initially done in the context of Wiles's solution to Fermat's Last Theorem.

I'm not sure what you're confused about in the statement of the result (which is correct). The Eichler-Shimura construction associated to any weight $2$ newform $f$ on $\Gamma_0(N)$ with Fourier coefficients in $\mathbb{Q}$ an elliptic curve $E_f/\mathbb{Q}$ with conductor $N$. The construction at the $\mathbb{C}$-analytic level is given in Dietrich Burde's answer: otherwise put, in the Jacobian variety you impose the relations $T_n(f) = a_n(f)$ for every positive integer $n$, so this is a kind of souped-up eigenspace construction. The fact that the elliptic curve is defined over $\mathbb{Q}$ is not trivial and the fact that it has conductor precisely $N$ (the level of $f$) is really nontrivial. The converse, namely that every rational elliptic curve of conductor $N$ is isogenous to a unique curve $E_f$ is the above result, which is a plausible candidate for the deepest theorem of late 20th century number theory.

Answer 2

Theorem $2$ is a very deep theorem saying that all elliptic curves over $\mathbb{Q}$ are modular. This was formerly known as the Modularity conjecture, and was solved by Wiles (and Breuil, Diamond, Conrad, Taylor) in the context of Fermat's last theorem. The point is that the map from rational newforms of level $N$ to isogeny classes of elliptic curves over $\mathbb{Q}$ of conductor $N$, $f\to E_f$, is really surjective.

Edit: The association $f \mapsto E_f$ is due to Shimura and Eichler. In the book of A. Knapp, Elliptic curves, it is explained how this map is constructed (basically to a rational newform $f$ an elliptic curve $E_f=\mathbb{C}/\Lambda_f$ is constructed, where $\Lambda_f$ is a certain lattice of periods; and it is a deep result that this elliptic curve is really defined over $\mathbb{Q}$).