Modularity of Elliptic Curves over $\mathbf {Q}$

Fix an elliptic curve $E$ over  $\mathbf {Q}$. In this section we will explain what it means for $E$ to be modular, and note the connection with Conjecture 10.2.3 from the previous section.

First, we give the general definition of modular form (of weight $2$). The complex upper half plane is $\mathfrak{h}= \{z \in \mathbf{C}:$   Im$(z) > 0\}.$ A cuspidal modular form $f$ of level $N$ (of weight $2$) is a holomorphic function $f : \mathfrak{h}\to \mathbf{C}$ such that $\lim_{z\to i\infty} f(z) = 0$ and for every integer matrix $\left( \begin{smallmatrix}a&b c&d\end{smallmatrix}\right)$ with determinant $1$ and $c\equiv 0 \pmod{N}$, we have

$$f\left( \frac{az + b}{cz + d} \right) = (cz+d)^{-2} f(z).$$

For each prime number $\ell$ of good reduction, let $a_\ell = \ell+1 - \char93 \tilde{E}(\mathbf{F}_{\ell})$. If $\ell$ is a prime of bad reduction let $a_\ell = 0,1,-1$, depending on how singular the reduction $\tilde{E}$ of $E$ is over $\mathbf{F}_{\ell}$. If $\tilde{E}$ has a cusp, then $a_\ell=0$, and $a_\ell=1$ or $-1$ if $\tilde{E}$ has a node; in particular, let $a_\ell=1$ if and only if the tangents at the cusp are defined over  $\mathbf{F}_{\ell}$.

Extend the definition of the $a_\ell$ to $a_n$ for all positive integers $n$ as follows. If $\gcd(n,m)=1$ let $a_{nm} = a_n \cdot a_m$. If $p^r$ is a power of a prime $p$ of good reduction, let

$$a_{p^r} = a_{p^{r-1}}\cdot a_p - p \cdot a_{p^{r-2}}.$$

If $p$ is a prime of bad reduction let $a_{p^r} = (a_p)^r$.

Attach to $E$ the function

$$f_E(z) = \sum_{n=1}^{\infty} a_n e^{2\pi i z}.$$

It is an extremely deep theorem that $f_E(z)$ is actually a cuspidal modular form, and not just some random function.

The following theorem is called the modularity theorem for elliptic curves over  $\mathbf {Q}$. Before it was proved it was known as the Taniyama-Shimura-Weil conjecture.

Theorem 10.2.4 (Wiles, Brueil, Conrad, Diamond, Taylor)   Every elliptic curve over $\mathbf {Q}$ is modular, i.e, the function $f_E(z)$ is a cuspidal modular form.

Corollary 10.2.5 (Hecke)   If $E$ is an elliptic curve over  $\mathbf {Q}$, then the $L$-function $L(E,s)$ has an analytic continuous to the whole complex plane.