# What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).

Let me first of all recall that a curve $C$ over a function field can be recovered from its Jacobian $Pic(C)$ and its theta divisor $\Theta \subset Pic^{g-1}(C)$ ($g$ the genus of $C$).

In their article van der Geer and Schoof construct analogues of the Jacobian and the theta divisor in the case of number fields.

Namely they consider the Arakelov-Picard group $Pic(F)$ of a number field $F$ and define its theta divisor as restriction of the function $h^0$ on $Pic(F)$ given by $$h^0(D) = log(\sum_{f \in I} e^{-\pi ||f||^2_D}),$$ where $I$ is a lattice associated with the Arakelov divisor $D$, to the subspace $Pic^{(d)}(F)$ of Arakelov divisors of degree $d$. Here $d$ is a suitable analogue of the genus of a curve.

Now they say that "it should be possible to reconstruct the arithmetic of the number field $F$ from $Pic^{(d)}(F)$ together with $h^0$".

My question is now of course which parts of the arithmetic of $F$ are known to be recovered from their analogy? For example I would be very interested in the question if the units of $F$ can be recovered somehow.

• Anyway, the purpose is solved. You came to know about the paper I have kept . I am happy. I am deleting my answer. Why to welcome troubles unnecessarily. Let peace prevail. – Shanmukha_Srinivasan Aug 6 '12 at 17:28
• Since I think it is useful information to have around, I summarize from memory from the deleted answer and subsequent discussion: the above mentioned paper is the thesis of R. Groenewegen "Vector bundles and geometry of numbers", more specifically its second part "Torelli for number fields", which explores the problem mentioned in the question, obtaining interesting results related to it, yet not answering the question itself. (I hope this is an accurate and neutral summary of the situation.) The document itself is available eg here: math.leidenuniv.nl/en/theses/13 – user9072 Aug 6 '12 at 17:49