On Shimura lifting of Hilbert modular forms

摘要

Let $${mathfrak {N}}_{0}$$ 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 $${mathfrak {N}}_{0}$$ 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 $$16|{mathfrak {N}}_{0}$$ 16 | 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 $$K={mathbf {Q}}(sqrt{2})$$ K = Q ( 2 ) and $$K={mathbf {Q}}(sqrt{5})$$ K = Q ( 5 ) , and obtain the formulas for the numbers of representations of totally positive integers in K as sums of three integral squares.