Integral points of bounded height on compactifications of semi-simple groups (Q2856613)

The main result of the paper is an asymptotic formula for the number of \(S\)-integral points of bounded height on partial bi-equivariant compactifications of a split semi-simple group of adjoint type \(G\) over a number field \(F\).NEWLINENEWLINETo be slightly more explicit, let \(X\) be the wonderful compactification of \(G\), if \(\lambda\) is an element of the Picard group of \(X\), there is a Weil height function on \(X\) with respect to \(\lambda\), \(H(\lambda,\;)\). Let \(D\) be a divisor in \(X\setminus G\) and let \(S\) be a finite set of places of \(F\) containing the Archimedean places. There is a notion of \((D,S)\)-integrality for points on \(X\) which agrees with \(S\)-integrality for points on a model of \(X\setminus D\) over the integers.NEWLINENEWLINEFix \(\lambda\) corresponding to a big line bundle on \(X\), the number of \((D,S)\)-integral points \(\gamma\) on \(G(F)\) such that \(H(\lambda,\gamma)\leq B\), for \(B\) going to infinity, is equal to NEWLINE\[NEWLINEc\,B^a\,\log(B)^{b-1}(1+o(1)),NEWLINE\]NEWLINE where \(a\), \(b\) and \(c\) are explicit constants depending only on the data \(X\), \(\lambda\), \(D\) and \(S\).