关于非同余数的一些结论（英文）

Some results on non-congruent numbers

2002年,冯克勤给出了一些与同余数相关的椭圆曲线Selmer群平凡的充要条件,通过这些条件可以证明关于非同余数的一些已有结论,还可以得到一些新的非同余数.本文通过类似方法计算了此类椭圆曲线的弱Mordell-Weil群,得到了一些使得该群平凡的条件,利用这些方法和条件同样可以得到冯的结果,并且可以找到一些新的非同余数.

In 2002, Feng gave some necessary and sufficient conditions on the vanishing of the Selmer groups of congruent elliptic curves. In virtue of these conditions, many old results on non-congruent numbers can be easily proved and more new non-congruent numbers can be found. In this paper, the au- thor will use the similar methods to compute the weak Mordell-Weil groups of those elliptic curves and get some necessary conditions on the vanishing of this group. Moreover, we can also use our method to prove some known results, and exemplify the efficiency of our conditions by some new non-congruent numbers.