On Zagier's adele

Don Zagier suggested a natural construction, which associates a real number and $p$-adic numbers for all primes $p$ to the cusp form $g=\Delta$ of weight $12$. 

He claimed that these quantities constitute a rational adele. I will discuss the ideas behind a proof of a similar statement when $g$ is a weight $2$ primitive 

form with rational integer Fourier coefficients. This proof makes use of a version of the Hodge decomposition for the formal group law of the rational elliptic 

curve associated with $g$, and boils down to an elementary observation on the integrality of the classical addition law of Weierstrass $\zeta$-function.