椭圆曲线Weierstrass模型的局部和全局密度

IF 0.6 3区 数学 Q3 MATHEMATICS
J. E. Cremona, M. Sadek
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引用次数: 25

摘要

我们证明了$\mathbb{Q}_p$上具有不同约简类型的椭圆曲线的$p$ -进密度的局部结果,以及$\mathbb{Q}$上具有指定约简类型的椭圆曲线在一个或多个(包括无穷多个)素数上的密度的全局结果。这些全局结果包括:在$\mathbb{Q}$上半稳定椭圆曲线(即具有无平方导体的椭圆曲线)的最小模型的积分Weierstrass方程的密度为$1/\zeta(2)\approx60.79\%$,与无平方整数的密度相同;$\mathbb{Q}$上半稳定椭圆曲线的密度为$\zeta(10)/\zeta(2)\approx60.85\%$;具有无平方判别式的Weierstrass积分方程的密度为$\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right) \approx 42.89\%$,与具有无平方判别式的monic积分三次多项式的密度相同(除了质数$2$处有不同的因子)(并且与Baier和Browning先前关于短Weierstrass方程的结果一致);在$\mathbb{Q}$上具有最小二乘判别式的椭圆曲线密度为$\zeta(10)\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right)\approx42.93\%$。局部结果来源于对Tate算法的详细分析,而全局结果则是通过使用由Poonen、Stoll和Bhargava开发的Ekedahl筛获得的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Local and global densities for Weierstrass models of elliptic curves
We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/\zeta(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\mathbb{Q}$ is $\zeta(10)/\zeta(2)\approx60.85\%$; the density of integral Weierstrass equations which have square-free discriminant is $\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right) \approx 42.89\%$, which is the same (except for a different factor at the prime $2$) as the density of monic integral cubic polynomials with square-free discriminant (and agrees with a previous result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over $\mathbb{Q}$ with square-free minimal discriminant is $\zeta(10)\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right)\approx42.93\%$. The local results derive from a detailed analysis of Tate's Algorithm, while the global ones are obtained through the use of the Ekedahl Sieve, as developed by Poonen, Stoll, and Bhargava.
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
9
审稿时长
6.0 months
期刊介绍: Dedicated to publication of complete and important papers of original research in all areas of mathematics. Expository papers and research announcements of exceptional interest are also occasionally published. High standards are applied in evaluating submissions; the entire editorial board must approve the acceptance of any paper.
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