When describing smooth algebraic curves over a field $k$, there are (at least) two useful notions of "class group". The first generalizes easily to general schemes: the Picard group can be defined as the group of Cartier divisors modulo linear equivalence, and for the ring of integers $\mathcal{O}_K$ of a number field, the Picard group is the ideal class group of fractional ideals modulo principal ideals.
However, there is a difference: the ideal class group of a number field is always finite, while the class group of a smooth projective curve $X$ is infinite, and classified by degree. This is defined to be (at least when $k = \overline{k}$ is algebraically closed) the sum of the multiplicities of a divisor $D$ at each closed point $x \in k$. It's well-defined on ideal classes due to the crucial property that $\deg (f) = 0$ for $(f)$ a principal divisor. The subgroup generated by elements of degree $0$ is the invariant $\text{Pic}^0(X)$, which is actually an abelian variety of dimension equal to the genus of the curve.
Why are these two types of behavior so different? Is there a way to define a notion of degree and $\text{Pic}^0$ for a number field $K$? One fact that seems promising is the product formula $\prod_\nu |x|_\nu = 1$ where $\nu$ runs through all places of $K$, both finite and infinite. If the extra infinite places are supposed to be analogous to the points at infinity of a projective curve, can we extend this analogy to define the degree of a fractional ideal? While the valuation at the finite places clearly can be extended, it's unclear what to do about the infinite places here.