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.