# Relationship between $\theta$ functions and number fields.

I'm trying to have a clear picture of the relationship of theta functions and $$L$$-functions, and the geometric objects they relate to.

Firstly, I know that $$\theta$$-functions arise as sections of line bundles on abelian varieties. So if you take a lattice $$\Lambda$$ in $$V=\mathbb{C}^g$$ and a polarization $$H$$ (i.e a hermitian form on $$V$$ such that $$E=Im(H)$$ is definite positive and integral on $$\Lambda$$) you get an abelian variety $$X=V/\Lambda$$ and a class of line bundles $$L(H, \bullet)$$. If you specify a quasi-character of $$H$$, then you get a well definied line bundle $$L$$ on $$X$$ whose 1st chern class is $$E$$. So far this is clear.

From the data of $$\Lambda$$ alone, you get a family of $$\theta$$ functions, which correspond to all sections of all the line bundles on $$X$$. Basically $$\theta$$-functions are paramterized by a lattice, a polarization and a quasi-character. You can also parametrize them by an element in the Siegel half space, a polarization and a quasi-character.

Now on the other hand $$L$$-functions arise as Mellin Transform of $$\theta$$-functions (up to Euler/Gamma-factors). For instance the Riemann Zeta function satsifies $$\pi^{-s/2}\Gamma(s/2)\zeta(s)=Mel(\theta(it)-1, s)$$ with $$\theta(z)=\sum_{m\in \mathbb{Z}}e^{i\pi m^2z}$$ the Jacobi theta function.

Of course the term $$it$$ in $$Mel(\theta(it)-1, s)$$ makes me think that $$it$$ lives in $$\mathbb{H}$$ the upper half plane, but I don't think that should be relevant because the variable $$t$$ is the one integrated in the Mellin transform so it's not a fixed parameter that I could interpret as corresponding to some lattice.

So my (rather blurry) question is : what is the relationship between the "abelian variety" picture of theta functions, and the zeta functions of number field arrising as Mellin transforms of the same $$\theta$$-functions.

The naive picture I have in mind (which is probably not true) is the following.

If you take $$k$$ a number field, its ring of integer gives you via the canonnical embedding a lattice $$\Lambda$$ in some vector space $$\mathbb{C}^g$$ (obvisouly there are some problems already, as the dimension of $$\mathbb{R}^{s+t}$$ has no reason to be even in general). The abelian variety $$X=\mathbb{C}^g/\Lambda$$ has a polarization (coming from the trace on the number field ?) and thus a line bundle, unique up to translation which has a section defined by a $$\theta$$ function, such that $$\zeta_k$$ is essentially the Mellin transform of that $$\theta$$-function.

Now what is wrong with that picture? How far is it from what actually happens? And what is the real picture?

I've seen $$\theta$$-functions in the context of $$L$$-functions, parametrized by all sorts of parameters, e.g Neukirch writes $$\theta_\Gamma(a,b,z)=\sum_{g\in \Gamma}N(a+g)e^{i\pi((a+g)z, a+g)+2i\pi(b,g)}$$ in this case can I interpret $$\theta_\Gamma(a,b,z)$$ as a section on some line bundle on $$\mathbb{C}^d/\Gamma$$ ?

I feel like those $$(a,b)$$ should be some $$(H,\alpha)$$ for some "natural" $$(H,\alpha)$$, am I completely mistaken here?

• Can you expand on the $\theta$ you wish to associate to $\Bbb{C}^g/\Lambda$ ? For non quadratic-imaginary number fields we need $\theta$ functions of several variable, change of variable $(t_1,\ldots,t_N) = (e^{v_1},\ldots,e^{v_N})$ the "Mellin transform" is $f(s)= \int_{\Bbb{R}^N/U} (\prod_{n=1}^N e^{v_n s}) \theta(i e^{v_1},\ldots,i e^{v_N}) d^N v$ where $U=\sum_{j=1}^J \Bbb{Z} \log u_j$ is the image of the unit group $O_k^\times$ and quotienting by $U$ is intended to kill $O_k^\times$. – reuns Jun 10 at 21:39
• For Neukrich the $a$ variable is intended to be set to $0$, it is there for the functional equation in the same way that it is the Fourier series of $\vartheta_t(a)=\sum_n e^{-\pi t(a+n)^2} = \sum_k t^{-1/2}e^{-\pi k^2/t} e^{2i \pi ak}$ which tells us the functional equation of $\pi^{-s/2}\Gamma(s/2)\zeta(s) = \frac12\int_0^\infty t^{s/2-1}(\vartheta_t(0)-1)dt$. – reuns Jun 10 at 21:43
• Well I wouldn't know exactly what $\theta$ I would want to associate, I'm thinking something like a section of a line bundle on $\mathbb{C}^g/j(O_k)$ where $j$ is the canonical embedding. – Elmoco Jun 11 at 8:06
• What baffles me is that these $\theta$-functions which are associated to number fields via their zeta functions, look a lot like the theta functions associated to line bundles on abelian variety, and I feel like there should be some connection between the two of them. But maybe they're not really related and I should only consider $\theta$-functions for number fields as ad-hoc functions? – Elmoco Jun 11 at 8:12
• In the case of imaginary quadratic fields, can we say something? I mean in this case $O_k$ is a lattice in $\mathbb{C}$ and thus defines a CM elliptic curve $E/\mathbb{C}$. Can we find a line bundle on it, such that $\zeta_k(s)$ is naturally $Mel(\theta, s)$ for $\theta$ defining a section of this line bundle. – Elmoco Jun 11 at 12:50