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

Abstract

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 ${\Phi }_{u,v}:E(\mathbb{Q})\to \mathrm{CL}(-D)$ 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 $\epsilon >0$, we show that as $X\to \infty$, we have $$\#\left\{-X<-D<0:-D\in {\Psi }_E\right\}{\gg }_{\epsilon }{X}^{\frac{1}{2}-\epsilon }.$$ 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 ${\gg }_{\epsilon }{X}^{\frac{1}{2}-\epsilon }.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.

17 pages, 1 table