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 $1/2$?


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...

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.