Integral division points on curves (Q2873597)

Let \(k\) be a number field and \(S\) a set of primes containing all the infinite ones. Let \(A/k\) be a semi-abelian variety, \(\Gamma_0\) a finitely generated subgroup of \(A(\overline k)\) and \(\Gamma \subseteq A(\overline k)\) be the division group attached to \(\Gamma_0\), i.e the set of points \(P\in A(\overline k)\) such that there exists a integer \(n\) such that \(nP\in \Gamma_0\).NEWLINENEWLINEIf \(X/k\) is any variety and \(\overline X\) its completion, define \(\partial X:=\overline X - X\). Let \(T\) be any subset of \(\overline X\), and let \(\overline T\) be its Zariski closure in \(\overline X\). Any \(P\in X(\overline k)\) is said to be \(S\)-integral relative to \(T\) if it is \((\overline T \cup \partial X, S)\)-integral on \(\overline X\).NEWLINENEWLINEThe authors pose the following conjecture:NEWLINENEWLINE\textit{Conjecture} Let \(k\) and \(S\) be as above and let \(A/k\) be a semi-abelian variety and \(\Gamma\) a division group in \(A(\overline k)\). Suppose that \(D\) is a non-zero effective divisor on \(A\) which is not the translate of any torsion divisor by any point of \(\Gamma\). Then the set NEWLINE\[NEWLINE\{P\in \Gamma: P \text{ is \(S\)-integral relative to }D\}NEWLINE\]NEWLINE is not Zariski dense in \(A\).NEWLINENEWLINEThe authors then prove the conjecture for \(1\)-dimensional semi-abelian varieties, i.e. for elliptic curves and \(1\)-dimensional tori.