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

TL;DR
This paper proves that for large enough k coprime to a prime , Erd53s-Selfridge curves have only trivial rational points, advancing understanding of Sander's conjecture using combinatorial and arithmetic techniques.
Contribution
It establishes new cases where Erd53s-Selfridge curves lack nontrivial rational points, introducing a novel mass increment method inspired by additive combinatorics.
Findings
Proves triviality of rational points for large coprime k and prime .
Introduces a new mass increment technique based on Faltings's theorem.
Advances partial resolution of Sander's conjecture.
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 are integers. These curves contain "trivial" rational points with , and a conjecture of Sander predicts for which pairs the curve contains "nontrivial" rational points where . Suppose is a prime. We prove that if is sufficiently large and coprime to , 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…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Vietnamese History and Culture Studies · Advanced Numerical Analysis Techniques
