Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms

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Abstract

The conjecture is made that the rational structures on spaces of modular forms coming from the rationality of Fourier coefficients and the rationality of periods are not compatible. A consequence would be that ζ(2k-1)/π 2k-1 (ζ(s)=Riemann zeta function;k∈ℕ,k≥2) is irrational or even transcendental.