# Theta function of an ideal

While attempting to find the functional equation for the dedekind zeta function, I encountered this function: $$\theta(x_1,x_2,...,x_{r_1+r_2})=\sum_{\alpha \in I_c} e^{-\pi \sum_{k=1}^{r_1+r_2}x_k \mid \sigma_k(\alpha) \mid^2}$$ Where $$I_c$$ is an ideal in a class group, $$\sigma_k$$ are the embeddings of the field, and $$r_1+r_2$$ is the rank of the unit group+1. Does this function obey a functional equation? Or is there a similar function that does? Thanks for any help.

## 1 Answer

$$I_c$$ is a lattice. Fix $$a_j$$ then for some matrix $$M = M(a,c)$$ $$\theta(a_1 t,\ldots a_n t) = \sum_{l \in \Bbb{Z}^n} e^{- \pi t \| Ml \|^2} = \sum_{k \in \Bbb{Z}^n} t^{-1} e^{- \pi t^{-1}\| M^{-\top} k \|^2}$$ (Poisson summation formula, Fourier series, Fourier transform of the Gaussian..)

The functional equation of $$\zeta_K(s)$$ is obtained from that $$M^{-\top}$$ corresponds to what we'd get when replacing $$(a_1,\ldots,a_n), I_c$$ by $$(a_1^{-1},\ldots,a_n^{-1}), I_{c^{-1}}$$

• What would that matrix look like? – uhhhhidk Jun 28 at 23:48