On the representability of Hilbert cusp forms by theta series

Using trace formulas for Hecke operators, Eichler first provided a positive solution about basis problems of elliptic cusp forms by quadratic forms. J-L. Waldspurger established that elliptic cusp forms of arbitrary level are spanned by theta series by means of different and interesting ideas and methods. This result is given by Zagier’s analytic theorems, the Siegel main theorem of quadratic forms and the theory of Hecke operators. We intend to generalize Waldspurger’s results and determine theta series which span the space of Hilbert new forms over arbitrary totally real algebraic number fields following Waldspurger’s methods.