Elliptic curves and lower bounds for class numbers

Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log D)^{\frac{r}{2}}$$ for sufficiently large discriminants $-D$ in certain families, where $c(E)$ is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants.