On the Fourier coefficients of Hilbert–Maass wave forms of half integral weight over arbitrary algebraic number fields

Abstract

The purpose of this paper is to derive a generalization of Shimura's results concerning Fourier coefficients of Hilbert modular forms of half integral weight over total real number fields in the case of Hilbert–Maass wave forms over algebraic number fields by following the Shimura's method. Employing theta functions, we shall construct the Shimura correspondence Ψ τ from Hilbert–Maass wave forms f of half integral weight over algebraic number fields to Hilbert–Maass wave forms Ψ τ( f) of integral weight over algebraic number fields. We shall determine explicitly the Fourier coefficients of Ψ τ( f) in terms of these f. Moreover, under some assumptions about f concerning the multiplicity one theorem with respect to Hecke operators, we shall establish an explicit connection between the square of Fourier coefficients of f and the central value of quadratic twisted L-series associated with the image Ψ τ( f) of f.