TITLE: Examples of Dedekind rings with infinite class number
QUESTION [5 upvotes]: I am looking for explicit examples of Dedekind rings with infinite class number. In most books on algebraic number theory there is a standard example (before or after proving that the class number is finite for the ring of integers in a number field), namely
$$
\mathbb{C}[x,y]/(y^2-x^3-ax-b),
$$
for affine elliptic curves $y^2=x^3+ax+b$, i.e., with $-4a^3-27b^2\neq 0$. 
Are there other explicit examples, possibly even with a nice proof that the class number is infinite ?

REPLY [8 votes]: Generalizing your example, we have the following:

If $X/\mathbf C$ is any smooth projective curve of genus $g>0$, and
  $U=\text{Spec }A$ is any open affine of it, then $A$ is a Dedekind
  ring whose Picard group has the cardinality of the continuum.

The reason is that $Pic(X)$ is (up to choice of a base point) canonically isomorphic to $\mathbf Z\times Pic(X)_0$, where $Pic(X)_0$ is the group of $\mathbf C$-points of an abelian variety of dimension $g$ over $\mathbf C$.
When passing from $X$ to $U$, the effect on the Picard group is essentially that of killing the divisor classes with support in $X-U$. There are finitely many of those, so $Pic(X)$ and $Pic(U)$ differ by a finitely-generated abelian group; in particular, they have the same cardinality.