Nontrivial rational points on Erd\H{o}s-Selfridge curves

Abstract

We study rational points on the Erd\H{o}s-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 Erd\H{o}s-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 Erd\H{o}s, 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.