اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn Class Numbers, Torsion Subgroups, and Quadratic Twists of Elliptic Curves
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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 $$#, left{-X < -D < 0: -D in Psi_E right } , gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$$ 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.
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