It is excerpt from "Algebraic Geometry Codes Basic Notions"(<https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCoQFjAA&url=http%3A%2F%2Fwww.math.umass.edu%2F~hajir%2Fm499c%2Ftvn-book.pdf&ei=CtLiUc2aAums4ATDloGoBA&usg=AFQjCNH8m6i46UGeRvF8J0nV_cMriSYSww&sig2=-doWN37rrQ2BMFnyUj3c1g&bvm=bv.48705608,d.bGE&cad=rjt>) pg 190: 
"The fractional ideal a of the ﬁeld $\mathbb{K}$ is the same divisor $D$ written multiplicatively. Then the space $L(D)$ corresponds to a
$a^{-1}$. The ideal class group
$Cl\_{K}$ is almost the group of $\mathbb{F}\_r$-points of the Jacobian of the curve; itscardinality
corresponds in fact to the product of the class number of the number ﬁeld by its regulator."


So I want to understand accurate statement. Is it true that for every curve $C$ over finite field there is number field $\mathbb{K}$ that Jacobian $C$ is isomorphic $Cl\_{\mathbb{K}}$?


If is it true, how to build it number field for some curve?



Analogy between Jacobian of curve and Ideal class group
I think this excerpt from Tsfasman-Vladut-Nogin should not be taken as a literal statement--rather it is an explication of (part of) the number field/function field dictionary. Most of this dictionary comes from the fact that these fields are the function fields of Dedekind schemes.


In particular, suppose $X$ is a Dedekind scheme, with function field $K$ (this may be either a function field or a number field). If $X$ is a projective curve over a field, rational points of its Jacobian correspond to degree $0$ line bundles $\mathcal{L}$ on $X$; a choice of meromorphic section of $\mathcal{L}$ embeds $\mathcal{L}$ into the constant sheaf $\underline{K}$ on $X$. On any affine open $\operatorname{Spec} \mathcal{O}$, this lets one view $\mathcal{L}$ as a fractional ideal. Conversely, fractional ideals for a Dedekind subring $R$ of $K$ correspond to line bundles (locally free sheaves of rank one) on $\operatorname{Spec} R$; that is, line bundles defined on open subsets of $X$.


That is, you should not think of this dictionary as assigning to each function field a number field; rather, it reinterprets the algebraic notion of a fractional ideal as a geometric notion--a line bundle--in the function field case. This analogy goes quite far; for example, if $X$ is a curve over a finite field $\mathbb{F}\_q$, with zeta function $\zeta\_X(t)$, one has


$$\operatorname{res}\_{t=1} \zeta\_X(t)=\frac{\#|\operatorname{Jac} X(\mathbb{F}\_q)|}{1-q}$$


which is a function field analogue of the [analytic class number formula](http://en.wikipedia.org/wiki/Class_number_formula). Here the numerator counts line bundles and the denominator counts units; this analogy (and in particular the lack of a regulator term) explains why Tsafaman-Vladut-Nogin refer to the rational points of the Jacobian as corresponding to the size of the class group times the regulator.


