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 $\mathrm {CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in \mathbb {Z}^2$, we define a family of homomorphisms $\Phi _{u,v}: E(\mathbb {Q}) \rightarrow \mathrm {CL}(-D)$ 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 $\Psi _E$ 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 $\mathrm {CL}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E) \log (D)^{\frac {r}{2}}$ as $D \to \infty$. Then for any $\varepsilon > 0$, we show that as $X \to \infty$, we have \begin{equation*} \#\, \left \{-X < -D < 0: -D \in \Psi _E \right \} \, \gg _{\varepsilon } X^{\frac {1}{2}-\varepsilon }. \end{equation*} In particular, if $\ell \in \{3,5,7\}$ and $\ell \mid |E_{\mathrm {tor}}(\mathbb {Q})|$, then the number of such discriminants $-D$ for which $\ell \mid h(-D)$ 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.