Generalization of a theorem of Hurwitz

This paper is an exposition of several classical results formulated and unified using more modern terminology. We generalize a classical theorem of Hurwitz and prove the following: let G(k) (z) = Sigma(m,n)' 1/(mz + n)(k) be the Eisenstein series of weight k attached to the full modular group. Let z be a CMpoint in the upper half-plane. Then there is a transcendental number Omega(z) such that G(2k) (z) = Omega(2k)(z) . (an algebraic number). Moreover, Omega(z) can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with algebraic Fourier coefficients, we prove that f (z) pi(k)/Omega(k)(z) is algebraic for any CM point z lying in the upper half-plane. We also prove that for any automorphism sigma of Gal((Q) over bar /Q), (f (z)pi(k)/Omega(k)(z))(sigma) = f(sigma)(z)pi(k)/Omega(k)(z).