Analogy between Jacobian of curve and Ideal class group 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?
 A: 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.  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.
