Suppose $F/\mathbb{Q}$ is a totally real field of degree $d$ and class number one. Fix an ordering $\sigma\_1, \dots, \sigma\_d$ on the embeddings of $F$. Is


$\sum\_{\alpha \in \mathcal{O}\_F}e^{2\pi i (z\_1 \sigma\_1 (\alpha^2)+ \dots +z\_d \sigma\_d(\alpha^2))}$


a Hilbert modular form of parallel weight $1/2$? 



Jacobi's theta function over totally real fields
Yep -- though I have never thought through any technicalities regarding definition of half-integral weight Hilbert modular forms; I'm comfortable saying, at least, that the square of that theta function is a Hilbert modular form of weight 1. Harvey Cohn wrote several papers about this: see e.g.


MR0113855 (22 #4686)
Cohn, Harvey
Decomposition into four integral squares in the fields of $2^{1/2}$ and $3^{1/2}$.
Amer. J. Math. 82 1960 301--322. 


I advised a senior thesis student at Princeton, Jorge Cisneros, who wrote a very nice thesis working out representations by sums of four squares for Q(sqrt(7)); in this case, there's a cusp form in the relevant space of Hilbert modular forms so the "error" betweeen the number of representations and the relevant divisor sum forms a very nice distribution... 


