Nontrivial rational points on Erdős-Selfridge curves

Abstract

We study rational points on the Erdős-Selfridge curves \begin{align*} y^\ell = x(x+1)\cdots (x+k-1), \end{align*} where $k,\ell\geq 2$ are integers. These curves contain "trivial" rational points $(x,y)$ with $y=0$, and a conjecture of Sander predicts for which pairs $(k,\ell)$ the curve contains "nontrivial" rational points where $y\neq 0$. Suppose $\ell \geq 5$ is a prime. We prove that if $k$ is sufficiently large and coprime to $\ell$, then the corresponding Erdős-Selfridge curve contains only trivial rational points. This proves many cases of Sander's conjecture that were previously unknown. The proof relies on combinatorial ideas going back to Erdős, as well as a novel "mass increment argument" that is loosely inspired by increment arguments in additive combinatorics. The mass increment argument uses as its main arithmetic input a quantitative version of Faltings's theorem on rational points on curves of genus at least two.