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.