论文标题
椭圆曲线和班级编号的二次曲折
Quadratic twists of elliptic curves and class numbers
论文作者
论文摘要
对于正等级$ r $椭圆形曲线$ e（\ mathbb {q}）$，我们采用理想的班级配对 $ e（\ MATHBB {q}）\ times e _ { - d}（\ Mathbb {q}）\ rightArrow \ rightArrow \ Mathrm {cl}（ - d），$ $ $ $ for quadratic tist $ e _ { - d}（ - d}（ - d}（ - d}（ - d}）（\ mathbb {q}）$ with a $ y $ y $ y $ y $ y $ - 边界。对于曲线$ e^{（a）}：\ y^2 = x^3-a，$带等级$ r（a），$ this给出$$ h（-d）\ geq \ frac {1} {10} \ cdot \ frac {| e _ {\ Mathrm {tor}}}（\ Mathbb {q}）|} |} |} {\ sqrt {\ sqrt {r _ {r _ { （\ frac {r（a）} {2} +1 \ right）}} {（4π）^{\ frac {r（a）} {2}}} {2}} \ cdot \ cdot \ frac {\ log（log log（d）当$ r（a）\ geq 3 $时，戈德菲尔德，毛和Zagier的经典下限。我们证明，$ e _ { - d}^{（a）}（\ m athbb {q}）$具有这样的点（分别为平族cositure contienture下的$ \ geq 2 $）为$ \ ggg_ { 我们提供了无限的许多情况，其中$ r（a）\ geq 6 $。这些结果可以看作是Gouvêa和Mazur的经典估计值的类似物，即排名$ \ geq 2 $二次曲折的数量，此外，我们还可以对Goldfeld-Gross-Zagier类班级数量进行``log-power''改进。
For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height'' rational point, to obtain effective class number lower bounds. For the curves $E^{(a)}: \ y^2=x^3-a,$ with rank $r(a),$ this gives $$ h(-D) \geq \frac{1}{10}\cdot \frac{|E_{\mathrm{tor}}(\mathbb{Q})|}{\sqrt{R_{\mathbb{Q}}(E)}}\cdot \frac{Γ\left (\frac{r(a)}{2}+1\right)}{(4π)^{\frac{r(a)}{2}}} \cdot \frac{\log(D)^{\frac{r(a)}{2}}}{\log \log D}, $$ representing an improvement to the classical lower bound of Goldfeld, Gross and Zagier when $r(a)\geq 3$. We prove that the number of twists $E_{-D}^{(a)}(\mathbb{Q})$ with such a point (resp. with such a point and rank $\geq 2$ under the Parity Conjecture) is $\gg_{a,\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ We give infinitely many cases where $r(a)\geq 6$. These results can be viewed as an analogue of the classical estimate of Gouvêa and Mazur for the number of rank $\geq 2$ quadratic twists, where in addition we obtain ``log-power'' improvements to the Goldfeld-Gross-Zagier class number lower bound.
