4
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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 $O_k^\times$. $\endgroup$ – reuns Jun 10 at 21:39
  • $\begingroup$ 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 $\pi^{-s/2}\Gamma(s/2)\zeta(s) = \frac12\int_0^\infty t^{s/2-1}(\vartheta_t(0)-1)dt$. $\endgroup$ – reuns Jun 10 at 21:43
  • $\begingroup$ 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. $\endgroup$ – Elmoco Jun 11 at 8:06
  • $\begingroup$ 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? $\endgroup$ – Elmoco Jun 11 at 8:12
  • $\begingroup$ 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. $\endgroup$ – Elmoco Jun 11 at 12:50

Your Answer

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

Browse other questions tagged or ask your own question.