The distribution of $S$-integral points on ${\mathrm{SL}}_2$-orbit closures of binary forms

Abstract

We study the distribution of $S$-integral points on ${\mathrm{SL}}_2$-orbit closures of binary forms and prove an asymptotic formula for the number of $S$-integral points of bounded height on ${\mathrm{SL}}_2$-orbit closures of binary forms. This extends a result of Duke, Rudnick, and Sarnak. The main ingredients of the proof are the method of mixing developed by Eskin-McMullen and Benoist-Oh, Chambert-Loir-Tschinkel's study of asymptotic volume of height balls, and Hassett-Tschinkel's description of log resolutions of ${\mathrm{SL}}_2$-orbit closures of binary forms.

19 pages, some minor changes

References in corpus (3)

• Integral points of bounded height on partial equivariant compactifications of vector groups
• Integral points of bounded height on toric varieties
• A GIT Construction of Moduli Spaces of Stable Maps in Positive Characteristic

Cited by in corpus (2)

• Campana points of bounded height on vector group compactifications
• Asymptotic growth of translation-dilation orbits