Title: Elliptic curves and rational points in arithmetic progression Show Chinese title

Title: 椭圆曲线和有理点的算术级数

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve. We consider finite sequences of rational points $\{P_1,\ldots,P_N\}$ whose $x$-coordinates form an arithmetic progression in $\mathbb{Q}$. Under the assumption of Lang's conjecture on lower bounds for canonical height functions, we prove that the length $N$ of such sequences satisfies the upper bound $\ll A^r$, where $A$ is an absolute constant and $r$ is the Mordell-Weil rank of $E/\mathbb{Q}$. Furthermore, assuming the uniform boundedness of ranks of elliptic curves over $\mathbb{Q}$, the length $N$ satisfies a uniform upper bound independent of $E$. Show Chinese abstract

Abstract: 设$E/\mathbb{Q}$为一条椭圆曲线。 我们考虑有理点$\{P_1,\ldots,P_N\}$的有限序列，其中$x$坐标在$\mathbb{Q}$中形成一个等差数列。 在假设朗氏关于典范高度函数下界的猜想成立的情况下，我们证明了此类序列的长度$N$满足上界$\ll A^r$，其中$A$是一个绝对常数，$r$是$E/\mathbb{Q}$的莫德勒-韦伊秩。 此外，假设椭圆曲线在$\mathbb{Q}$上的秩的有界性，长度$N$满足一个与$E$无关的统一上界。