Journal of the Ramanujan Mathematical Society

Jung-Jo Lee ¹, M. Ram Murty ², Donghoon Park ³

Affiliations
1 Department of Mathematics, Kyungpook National University, Daegu-702-701, KR
2 Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario- K7L3N6, CA
3 Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul-120-749, KR

Abstract

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)=∑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^2k_z. (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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