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Generalization of a Theorem of Hurwitz
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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)=∑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 Ωz such that
G2k(z)=Ω2kz. (an algebraic number).
Moreover, Ωz 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)πk/Ωzk 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)πk/Ωzk)σ=fσ(z)πk/Ωzk.
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 Ωz such that
G2k(z)=Ω2kz. (an algebraic number).
Moreover, Ωz 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)πk/Ωzk 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)πk/Ωzk)σ=fσ(z)πk/Ωzk.
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