Elliptic curves and lower bounds for class numbers

Abstract

Ideal class pairings map the rational points of rank $r\ge 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D)\ge \frac{1}{2}(c(E)-\epsilon )(\log D{)}^{\frac{r}{2}}$$ 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.

Corrected a typo appearing in bounds