# 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?

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While I don't quite understand the first paragraph, the answer to the question, assuming you mean the rational points of the jacobian, then such a question is open. For example, we know how to construct an elliptic curve mod p with q points, q a prime close to p, but we do not know how to construct a number field with class group exactly of size q, for arbitrary q. – Dror Speiser Jul 14 '13 at 10:10

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