Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms (Q912901)

Let K be an algebraic number field and let \(M^ 0_{2k}\) be the K- vector space of modular forms with respect to SL(2,\({\mathbb{C}})\) whose Fourier coefficients are all in K. Let \(S^ 0_{2k}\) be the subspace of cusp forms. The author conjectures that for any \(f\in M^ 0_{2k}\) there is at least one even n with \(0\leq n\leq 2k-2\) such that \(L(f,n+1)/\pi^{n+1}\not\in K.\) Another conjecture is that for any \(f\in S^ 0_{2k}\) there is at least one odd n with \(0\leq n\leq 2k-2\) such that \(L(f,n+1)/\pi^{n+1}\not\in K.\) These conjectures are supported by a few known properties of periods of modular forms. A consequence would be that \(\zeta (2k-1)/\pi^{2k-1}\not\in \bar Q\) \((k=2,4,6,...)\).