Divisor class group and Picard group of a smooth curve ($\mathrm{Cl}(X)\cong \mathrm{Pic}(X)\cong H^1(X,{\mathcal{O}}_X^{\ast })$)

For a smooth irreducible algebraic curve, send a divisor on an algebraic curve $D={\sum }_P{n}_{P}P$ to the line bundle associated to a divisor defined locally by ${\mathcal{O}}_X(D{)}_P=t_P^{-n_P}{\mathcal{O}}_{X,P}$. A nonzero rational section of a line bundle gives its inverse construction, and changing that section changes its divisor by a principal divisor. Thus the divisor class group is the Picard group. The quotient of the sheaf of nonzero rational functions on an irreducible variety by the sheaf of units of the structure sheaf has stalk $\mathbb{Z}$ at each closed point, via the discrete valuation. Its global sections are finite sums of points. The corresponding long exact sequence in sheaf cohomology, and the vanishing for the rational-function sheaf, identify both groups with $H^1(X,{\mathcal{O}}_X^{\ast })$.

For an elliptic curve over an algebraically closed field, ${\mathrm{Pic}}^0(E)\cong E(k)$ via $P\mapsto [P-O]$. This group is infinite and divisible by any prime $\ell$ different from the characteristic, since $[\ell ]:E\to E$ has nonzero derivative and is a surjective morphism of projective curves. A finitely generated abelian group divisible by $\ell$ has zero free rank and is finite. Hence $E(k)$, and therefore $\mathrm{Cl}(E)$, cannot be finitely generated. This proof applies to countable algebraically closed fields as well.