Relation between elliptic curves and Dirichlet L-series

Question

I have read that an elliptic curve $E$ is modular if $a(n) = c(n)$ for all $n$, where $a(n)$ is the $n$-th coefficient in the Dirichlet series of $E$, $L(E,s)$, and $c(n)$ is the $n$-th coefficient in the Fourier series expansion of some modular form. I understand that, but how do I determine $a(n)$ and $L(E,s)$?

Answer 1

At a prime $p$ for which $E$ has good reduction, the coefficient $a(p)$ is just $p - 1 + \# E(\mathbb{F}_p)$, where $E(\mathbb{F}_p)$ is the set of points of $E$ modulo $p$. At the remaining primes, the definition is slightly more complicated. If $N$ is the conductor of $E$ and $p$ divides $N$ but $p^2$ does not divide $N$, then $E$ has multiplicative reduction modulo $p$ and we set $a(p) = 1$ if $E$ has split multiplicative reduction and $a(p) = -1$ if $E$ has nonsplit multiplicative reduction. If $p^2$ divides $N$, then $E$ has additive reduction modulo $p$ and we set $a(p) = 0$. We then define \[L_p(E,s) = \frac{1}{1 - a(p) p^{-s} + p^{1 - 2s}}\] at each good prime, and at the bad primes we set \[L_p(E,s) = \frac{1}{1 - a(p) p^{-s}}.\] We then define \[L(E,s) = \prod_p L_p(E,s).\] If you expand the Euler product, then you end up with a Dirichlet series \[\sum_{n = 1}^{\infty} \frac{a(n)}{n^s}\] whose coefficients $a(n)$ are multiplicative (but not completely multiplicative) with its value at primes $n = p$ equal to $a(p)$ defined as above.