On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients

The latter is a sort of 'L-indistinguishability' result concerning the (presumably transcendental) Petersson norms-squared of Hecke eigenfunctions. The general assertion is essential in discussion of special values of L-functions obtained as inner products. Incidental to the proof of this fact, we obtain a very short proof of the important (known) result that the space of holomorphic cuspforms of such weight and with respect to a principal congruence subyroup is spanned by those with rational Fourier coefficients (for weights as above). Rather than starting from the theory of canonical models, we begin with consideration of the arithmetic of the Fourier coefficients of Siegel's Eisenstein series: this is a relatively elementary issue, amenable to direct calculation (although, ironically, the best reference currently available seems to be [H3], wherein canonical models results are invoked). By contrast, previous proofs of results concerning rationality properties of Fourier coefficients have relied essentially upon the theory of canonical models: the paradigms are the two papers [Sh2] and [Sh3] of Shimura, which depend upon the sequence of his papers culminating in [Shl] which developed the necessary theory of canonical models.