论文标题
沃特金斯（Watkins）的椭圆形曲线的猜想
Watkins' conjecture for elliptic curves over function fields
论文作者
论文摘要
在2002年，沃特金斯（Watkins）猜想，鉴于椭圆曲线定义了$ \ mathbb {q} $，其mordell-weil等级最多是其模块化度的$ 2 $ adiC估值。我们考虑了积极特征的功能领域的类似问题，并且在某些情况下证明了这一点。更准确地说，每条模块化半稳定的椭圆曲线均超过$ \ mathbb {f} _q（t）$，在扩展了恒定的标量后，以及$ \ mathbb {f} _q（f} _q（t）$的每个二次旋转，由多态度由多个元素与wat soguils of watkins'conluce of watkins'conluine'contuins'contuins'''此外，对于由于乌尔默（Ulmer）而闻名的椭圆形曲线的著名家族，我们证明了沃特金斯（Watkins）的猜想的类似物。
In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over $\mathbb{F}_q(T)$ after extending constant scalars, and every quadratic twist of a modular elliptic curve over $\mathbb{F}_q(T)$ by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins' conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins' conjecture.
