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 = ∆ of weight 12. He claimed that these quantities constitute a rational adele. We give a simple modular proof of Zagier’s original claim, and prove a similar statement when g is a weight 2 primitive form with rational integer Fourier coefficients making use of a version of the Hodge decomposition for the formal group law of the rational elliptic curve associated with g. 0. Introduction Let g = ∑ n≥1 b(n)q ∈ Sk(N), with q = exp(2πiτ) and I(τ) > 0 be a primitive form of conductor N (i.e. a new normalized cusp Hecke eigenform on Γ0(N), cf. [11, Section 4.6]) of even integer weight k. Assume that all Fourier coefficients b(n) ∈ Z are rational integers. We denote by Eg the Eichler integral Eg(τ) = ∑