论文标题
椭圆曲线Weierstrass模型的本地和全球密度
Local and global densities for Weierstrass models of elliptic curves
论文作者
论文摘要
我们在$ \ mathbb {q} _p $上的椭圆曲线的$ p $ - 亚种密度上证明了本地结果，并具有不同的减排类型的全球结果，以及椭圆曲线的密度超过$ \ mathbb {q} $，具有指定的减少类型（包括无限的许多）总数。这些全局结果包括：积分WeierStrass方程的密度是$ \ Mathbb {Q} $的半椭圆曲线的最小模型（即带有无方形导体的椭圆形曲线）为$ 1/ζ（2）\ lift of 60.79 \％\％\％$，与平方五整数平方五整数的密度相同； $ \ mathbb {q} $是$ζ（10）/ζ（2）\ oft60.85 \％$;具有无平方判别性的积分WeierStrass方程的密度为$ \ prod_p \ left（1- \ frac {2} {2} {p^2}+\ frac {1} {1} {p^3} \ right）\约42.89 \％$，除了一个$ 2 $ $ 2 $）的元素（约有$ 2 $），该元素是$ 2 $的元素。无方判别（并同意拜尔和褐变的先前结果，用于简短的WeierStrass方程）；椭圆曲线的密度在$ \ mathbb {q} $上具有无平方的最小判别的密度为$ζ（10）\ prod_p \ left（1- \ frac {2} {p^2} {p^2}+\ frac {1} {1} {p^3} {p^3} \ right） 局部结果来自对泰特算法的详细分析，而全局的结果是通过使用Poonen，Stoll和Bhargava开发的Ekedahl筛子获得的。
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 $1/ζ(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\mathbb{Q}$ is $ζ(10)/ζ(2)\approx60.85\%$; 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 $ζ(10)\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right)\approx42.93\%$. 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.
