TITLE: Formal group laws and L-series
QUESTION [46 upvotes]: Let E be an elliptic curve, let $L(s) = \sum a_n n^{-s}$
denote its L-function, and set 
$$ f(x) = \sum a_n \frac{x^n}{n}. $$ 
Then Honda has observed that 
$$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$ 
defines a formal group law. 
The formal group law of an elliptic curve has applications to 
the theory of torsion points, apparently because formal groups
are useful tools for studying such objects over discrete valuation 
domains.
Nevertheless I would appreciate it if someone could point out
the intuition behind this approach. What is the connection
between the L-series and the group law on the curve given by
the formal group law? Do formal group laws just give a 
streamlined proof of basic properties of the elliptic curve 
over $p$-adic fields, or is there more to them? 
I've also seen the work of Lubin-Tate in local class field theory, and
I do remember that I found the material as frightening as cohomology 
at first. It would be nice if the answers had something from a salesman's
point of view: why should I buy formal group laws at all?

REPLY [49 votes]: 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.
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