Formal group laws and L-series

Okay, here's a few words about the relation between the $L$-series and the formal group. In general, if $F(X,Y)$ is the formal group law for $\hat G$, then there is an associated formal invariant differential $\omega(T)=P(T)dT$ given by $P(T)=F_X(0,T)^{-1}$. Formally integrating the power series $\omega(T)$ gives the formal logarithm $\ell(T)=\int_0^T\omega(T)$. The logarithm maps $\hat G$ to the additive formal group, so we can recover the formal group as $$ F(X,Y) = \ell^{-1}(\ell(X)+\ell(Y))$$. (See, e.g., Chapter IV of Arithmetic of Elliptic Curves for details.)

Now let $E$ be an elliptic curve and $\omega=dx/(2y+a_1x+a_3)$ be an invariant differential on $E$. If $E$ is modular, say corresponding to the cusp form $g(q)$, then we have (maybe up to a constant scaling factor) $\omega = g(q) dq/q = \sum_{n=1}^{\infty} a_nq^{n-1}$. Eichler-Shimura tell us that the coefficients of $g(q)$ are the coefficients of the $L$-series $L(s)=\sum_{n=1}^\infty a_n n^{-s}$. Integrating $\omega$ gives the elliptic logarithm, which is the function you denoted by $f$, i.e., $f(q)=\sum_{n=1}^\infty a_nq^n/n$, and then the formal group law on $E$ is $F(X,Y)=f^{-1}(f(X)+f(Y))$.

To me, the amazing thing here is that the Mellin transform of the invariant differential gives the $L$-series. Going from the invariant differential to the formal group law via the logarithm is quite natural.

The usual(?), or at least more prosaic, way to define the formal group law on an elliptic curve $E$ is to take a Weierstrass equation and expand the addition law in terms of the formal parameter $z = -x/y$, which is a uniformizer at the identity element of $E$. Then one studies formal groups abstractly, proves various properties, and uses them to deduce things about the elliptic curve. This is all spelled out in my book The Arithmetic of Elliptic Curves (Chapters IV and VII), as well as lots of other places. In particular, if $\hat G$ is a formal group over a ring $R$ of characteristic prime to $m$, then multiplication by $m$ is invertible. So if $R$ is a complete DVR with maximal ideal $M$ and residue characteristic $p$, then $\hat G(M)$ has no prime-to-$p$ torsion.

The reason this is important for elliptic curves is because we have an exact sequence in which the formal group is the kernel of the reduction mod $M$ map. So for example, if $E$ has good reduction modulo $M$, then there is an exact sequence $$ 0 \to \hat E(M) \to E(K) \to E(R/M) \to 0.$$ One can use this sequence to deduce ramification information about the fields generated by torsion points, and this in turn is a crucial ingredient in the proof of the (weak) Mordell-Weil theorem (op cit Chapter VIII).

So I guess my answer as to why you should "buy a formal group law" is that on an algebraic group $G$ over a (complete local) field $K$ with ring of integers $R$, maximal ideal $M$, and residue field $k$, the group $G(K)$ is complicated, so we can analyze it by breaking it up into the smaller group $G(k)$ and the formal group $\hat G(M)$. And formal groups are much easier to understand than algebraic groups.

I know this only partially answers your question, but the relation between this definition of the formal group of an elliptic curve and the coefficients of the $L$-series will have to wait for another post.