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?

Formal group laws and zeta-functions, Osaka J. Math, 5 (1968) 199-213. $\endgroup$