On Shimura lifting of Hilbert modular forms

Abstract

Let ${\mathfrak {N}}_{0}$ be a integral ideal divisible by 4, of a totally real field K. We show that there is the Shimura lifting map of a space of Hilbert modular forms with character modulo ${\mathfrak {N}}_{0}$ of half-integral weight, to the space of Hilbert modular forms of integral weight under some condition. In particular it is shown that if $16|{\mathfrak {N}}_{0}$, then any Hilbert modular forms of weight at least 5 / 2 has the Shimura lift. As an application, we compute the Shimura lifts of the third powers of theta series for $K={\mathbf {Q}}(\sqrt{2})$ and $K={\mathbf {Q}}(\sqrt{5})$, and obtain the formulas for the numbers of representations of totally positive integers in K as sums of three integral squares.