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

$\displaystyle 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

$\displaystyle 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

$\displaystyle 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.

William Stein 2012-09-24