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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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up vote 5 down vote accepted

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...

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Thanks, Jordan! – David Hansen Apr 28 '10 at 2:09

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