# Jacobi's theta function over totally real fields

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 $1/2$?

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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.