论文标题
使用2-DESCENT构建一致的椭圆曲线
Constructing congruent number elliptic curves using 2-descent
论文作者
论文摘要
一个积极的整数是某些理性右三角的区域，称为一个一致的数字。从代数角度来看，作为一个一致的数字意味着满足方程式系统。早在1800年代，就可以理解，如果$ n $是一个一致的数字，那么等式$ nm^2 = uv（u^2 -v^2）$具有$ \ mathbb {z} $中的解决方案。使用一致的数字和椭圆曲线之间的关系$ e_n：y^2 = x^3 -n^2 x $（在1900年代建立），我们将证明这两个世纪的结果的相反也成立。除此之外，我们还使用2-DESCENT方法提供了相反的另一个证明。在本文结尾处，我们演示了如何使用我们的证明来构建至少2和3的$ e_n $的亚家族。
A positive integer that is the area of some rational right triangle is called a congruent number. In an algebraic point of view, being a congruent number means satisfying a system of equations. As early as the 1800s, it is understood that if $n$ is a congruent number, then the equation $nm^2 = uv(u^2 - v^2)$ has a solution in $\mathbb{Z}$. Using the relation between congruent numbers and elliptic curves $E_n: y^2 = x^3 - n^2 x$ which was established in the 1900s, we will prove that the converse of this two century-old result holds as well. In addition to this, we present another proof of the converse using the method of 2-descent. Towards the end of this paper, we demonstrate how one can use our proof to construct subfamilies of $E_n$ with rank at least 2 and 3.
