arXiv 2006.010632020-06-27math.NT

椭圆曲线的二次扭曲与类数

Quadratic twists of elliptic curves and class numbers

Michael Griffin, Ken Ono, Wei-Lun Tsai

备注：: We correct minor typographical errors, including the formula in the abstract and equation (1.3)

摘要：

对于正秩的$r$椭圆曲线$E(\mathbb{Q})$，我们利用理想类对$$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$对二次扭曲$E_{-D}(\mathbb{Q})$和适当的“小$y$ -height”有理点，得到有效类数下界.对于具有秩$r(a),$的曲线$E^{(a)}: \ y^2=x^3-a,$，这给出了$$ h(-D) \geq \frac{1}{10}\cdot \frac{|E_{\mathrm{tor}}(\mathbb{Q})|}{\sqrt{R_{\mathbb{Q}}(E)}}\cdot \frac{Γ\left (\frac{r(a)}{2}+1\right)}{(4π)^{\frac{r(a)}{2}}} \cdot \frac{\log(D)^{\frac{r(a)}{2}}}{\log \log D}, $$，其表示当$r(a)\geq 3$时对Goldfeld、Gross和Zagier的经典下界的改进。我们证明了具有这样一个点的扭转数$E_{-D}^{(a)}(\mathbb{Q})$（分别为.在奇偶猜想下具有这样的点和秩的$\geq 2$）是$\gg_{a,\varepsilon} X^{\frac{1}{2}-\varepsilon}.$。这些结果可以被看作是一个模拟的经典估计的Gouvêa和Mazur的秩$\geq 2$二次扭曲的数量，此外，我们获得了“对数功率”的改进Golden-Gross-Zagier类数下限。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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