Computing the Picard group of Number Fields in analogy with genus $0$ curves It is possible to compute the Picard group of (some? all?) genus $0$ curves in the following manner:
For concreteness let, $A = k[x,y]/(x^2+y^2-1)$ and $X = \operatorname{Spec} A$. Let $Y$ denote $\operatorname{Proj} k[x,y,z]/(x^2+y^2-z^2)$. Let $k$ be any field for now.
Then, either by general theory or by explicit computation, we can find an isomorphism $\pi: Y \to \mathbb P^1$. Now $X$ is an open subscheme of $Y$ that either misses two degree one points or one degree two points depending on whether $k$ contains $\sqrt{-1}$.
By the above isomorphism, we know that $\operatorname{Pic} Y = \operatorname{Pic} \Bbb P^1 = \Bbb Z$. The Picard group is easily seen to be isomorphic to the Class groups in this case and the excision sequence gives us the following:
$$G \to \operatorname{Cl} Y \to \operatorname{Cl} X \to 0$$
where $G = \Bbb Z$ or $\Bbb Z\oplus \Bbb Z $ depending on whether $X$ misses one point or two. This lets us compute the class group of $X$ and it turns out to be either trivial or $\Bbb Z/2$.
The analogy:
I believe there is some sort of strong analogy between $A$ and $\Bbb Z[\sqrt{-5}]$ when $k$ does not contain $\sqrt{-1}$. For instance, I think both rings are not UFD's "only" because of the factorizations:
$$x^2 = (1-y)(1+y), 2\times 3 = (1+\sqrt{-5})(1+\sqrt{-5})$$
where $x$ is roughly equivalent to $2/3$ and $y$ is $\sqrt{-5}$. Both factorizations seem to occur because we can write a square as a difference of two squares($5 = 3^2-2^2$.)
Questions:
Can one make this analogy precise? Does it extend to other genus $0$ curves/(quadratic?) number fields? What about higher genus?
If one can make this analogy precise, does the above method also let us compute the Class groups of number fields?
A reference(with maybe a short explanation if possible) would be perfectly acceptable as an answer.
 A: Usually the analogy is drawn between number fields and curves over finite fields, concerning such vast and deep topics as CFT or L-functions. However you appear to ask for more than analogy in the specific case of genus zero. As stressed by Captain Lama, "computing the class group of a number field is something so ridiculously difficult" that your question looks like wishful thinking. But if you accept to abandon the « global » point of view for the « local » one, i.e. to look at phenomena at one single prime at a time, there may be more than an analogy. More precisely, fix a number field K and a prime number p (suppose p odd for simplicity), and denote by S the set of primes above p in K. Let $G_S(K)$ be the Galois group over K of its maximal pro-p-extension which is unramified outside S. One would like to give a description of $G_S(K)$  by generators and relations in the category of pro-p-groups. A minimal set of generators of $G_S(K)$ can be obtained using CFT and, roughly speaking, the $Z_p$-torsion submodule $T_S(K)$ of the abelianization of  $G_S(K)$ « contains » the relations. The field K is called p-rational if  $T_S(K)$ is zero, i.e. if $G_S(K)$ is pro-p-free. This is the exact analogue of the case of a curve X (projective, smooth, irreducible) over an algebraically closed field, where the role of $G_S(K)$ is played by the profinite fundamental group of X \ {${P_1, … , P_s , infty}$}, s being the cardinal of S . For details and developments, see Movahhedi & Nguyen Quang Do, « Sur l’arithmétique des corps de nombres p-rationnels », in Séminaire de Théorie des Nombres Paris 1987-88, Progress in Math., vol. 81, Birhäuser, especially pp. 157-158. The condition of p-rationality imposes of course conditions on the p-class group. If K is a quadratic imaginary field and p is at least 5, K is p-rational if  p does not divide theclass number of K. This is the case  of your example, but unfortunately, the information goes in the wrong sense !
