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?