$S$-parts of values of univariate polynomials,\\ binary forms and decomposable forms at integral points

Values of isotropic quadratic forms at S-integral points

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Values of Quadratic Forms at Integral Points : an Elementary Approach

Powerfree Values of Binary Forms

Given a binary form f(x; y) 2 Z[x; y], we will be interested in nding the smallest k for which we can establish that there are in nitely many integers a and b such that f(a; b) is k free. Necessarily, we require the f has no xed kth prime power divisor. Until the nal section of this paper, we will also consider f to be irreducible. We set n = deg f . For k = 2, this problem has recently become ...

The distribution of S-integral points on SL2-orbit closures of binary forms

We study the distribution of S-integral points on SL2-orbit closures of binary forms and prove an asymptotic formula for the number of S-integral points. 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-Tschin...

Values of Indefinite Quadratic Forms at Integral Points and Flows on Spaces of Lattices

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on closures of orbits in spaces of lattices such as SLn(R)/SLn(Z) . At first sight, A) belongs to analytic number theory and B) belongs to ergodic and Lie theory, a...