Heegner point constructions and fundamental units in cubic fields

We use Heegner points to prove the existence of nontorsion rational points on the elliptic curve $y^2=x^3+D$ for any rational number $D=a\mathrm{/}b$ such that $a$ and $b$ are squarefree integers for which $6$, $a$, and $b$ are pairwise relatively prime, $a\equiv b(\mathrm{m}\mathrm{o}\mathrm{d}4)$, $\mid a\mid \mid b{\mid }^{-1}\equiv 5$ or $7(\mathrm{m}\mathrm{o}\mathrm{d}9)$, and $h_K$ is odd, where $K:=\mathbb{Q}(\sqrt[3]{D})$. In particular, we show that under these assumptions, the elliptic curve with equation $y^2=x^3+D$ has algebraic rank $1$ and the elliptic curve with equation $y^2=x^3-D$ has algebraic rank $0$. This follows from our new expression for the fundamental unit of ${\mathcal{O}}_K$ in terms of the class number $h_K$ and the norm of a special value of a modular function of level $6$, for any integer $D$ relatively prime to $6$, not congruent to $\pm 1(\mathrm{m}\mathrm{o}\mathrm{d}9)$, for which no exponent in its prime factorization is a multiple of $3$. This expression is an analogue of a theorem of Dirichlet in 1840 relating the fundamental unit of a real quadratic field to its class number and a product of cyclotomic units.