On Some Arithmetic Properties of Modular Forms of One and Several Variables

Abstract

The purpose of this paper is twofold. We first present a principle that a certain non-holomorphic derivative of a Hilbert modular form with algebraic Fourier coefficients, divided by another modular form, takes an algebraic value at every point with 'complex multiplication'. Second, we investigate the Hilbert or Siegel modular forms with cyclotomic Fourier coefficients in the framework of canonical models as developed in our previous papers. The first principle, even specialized to the one-dimensional case, seems new and is as follows. Let f (z) and g(z) be modular forms of weight r and r + 2n, respectively, with respect to a congruence subgroup r of SL2(Z), and put