论文标题
二次椭圆表面的理性点
Rational points on quadratic elliptic surfaces
论文作者
论文摘要
我们认为椭圆表面的系数为$ 2 $ $ 2 $多项式在可变$ t $中。最近显示，对于无限的许多理性值$ t $，由此产生的椭圆曲线的排名至少为$ 1 $。在本文中，我们证明了每个这样的椭圆表面的Mordell-Weil等级最多为$ 6 $ $ \ Mathbb Q $。实际上，我们表明，这些椭圆表面的mordell-weil等级受到$ \ mathbb q $的某个多项式的零数的控制。
We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we prove that the Mordell-Weil rank of each such elliptic surface is at most $6$ over $\mathbb Q$. In fact, we show that the Mordell-Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over $\mathbb Q$.
