On Class Numbers, Torsion Subgroups, and Quadratic Twists of Elliptic Curves

The Mordell-Weil groups $E(\mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(\mathbb{Q})$ and the ideal class groups $\mathrm{C}\mathrm{L}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in $\mathbb{Z}$^2$,wedefineafamilyofhomomorphisms$\Phi_{u,v}: E($\mathbb{Q}$) \rightarrow $\mathrm{C}\mathrm{L}$(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let ${\mathrm{\Psi }}_E$ be the set of suitable fundamental discriminants $-D<0$ satisfying the following three conditions: the quadratic twist $E_{-D}$ has rank at least 1; $E_{\text{tor}}(\mathbb{Q})$ is a subgroup of $\mathrm{C}\mathrm{L}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E)\log (D{)}^{\frac{r}{2}}$ as $D\to \mathrm{\infty }$. Then for any $\epsilon >0$, we show that as $X\to \mathrm{\infty }$, we have $$\mathrm{\#}\{-X<-D<0:-D\in {\mathrm{\Psi }}_E\}{\gg }_{\epsilon }{X}^{\frac{1}{2}-\epsilon }\mathrm{.}$$ In particular, if $\mathrm{\ell }\in \{3,5,7\}$ and $\ell \mid |E_{$\mathrm{t}\mathrm{o}\mathrm{r}$}($\mathbb{Q}$)|$,thenthenumberofsuchdiscriminants$-D$forwhich$\ell \mid h(-D)$is$\gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.