In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined over the rational numbers or has complex multiplication.

中文翻译：

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椭圆曲线上具有相乘相关坐标的扭转点

在本文中，我们研究了椭圆曲线上坐标满足某些乘性相关关系的扭转点的有限性问题。特别是，我们证明了在一个数字字段上定义的椭圆曲线上，只有有限多个扭转点，其坐标与乘数相关。此外，当椭圆曲线在有理数上定义或具有复数乘法时，我们会产生有效的结果。