论文标题
椭圆曲线家族的根号和两个应用
Root numbers of a family of elliptic curves and two applications
论文作者
论文摘要
对于\ Mathbb {q} \ setMinus \ { - 1,0,1 \} $的每个$ t \，在$ \ mathbb {q} $上定义了椭圆曲线\ begin \ begin {align*} e_t：y^2 = x（x+1）（x+t^2）。 \ end {align*}使用root号$ w（e_t）$的公式作为$ t $的函数，并假设对椭圆曲线等级的一些标准猜想，我们确定（一组密度为零），一组椭圆形曲线的同态类别类别的椭圆形类别$ e/\ e/\ mathbb {q} $ nterm oft in n Times $ ntime oft in ofters n oft n Tims oft in n Times $ nterm oft in n Times $ nterbb contbb contbb in oft y MATHBB（ \ Mathbb {Z}/2 \ Mathbb {Z} \ Times \ Mathbb {Z}/4 \ Mathbb {Z} $，以及可以写入两个正理右三角形斜率的乘积的有理数。
For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves $E/\mathbb{Q}$ whose Mordell-Weil group contains $\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.
