Journal of the Ramanujan Mathematical Society

Volume 31, Issue 3, September 2016 pp. 215–226.

Generalization of a theorem of Hurwitz

Authors: Jung-Jo Lee, M. Ram Murty and Donghoon Park
Author institution:Department of Mathematics, Kyungpook National University, Daegu 702-701, South~Korea

Summary: 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 Gk(z)=∑ {m,n} {'}{1}/{(mz+n) {k}} be the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number Ω z such that G {2k}(z) = Ω z {2k} · (an algebraic number). Moreover, Ω 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)π k/Ω z k is algebraic for any CM point z lying in the upper half-plane. We also prove that for any automorphism σ of Gal(Q/Q), (f(z)π k/Ω z k) σ = f σ(z)π k/Ω z k.

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