The dynamical Mordell-Lang problem for étale maps.

*(English)*Zbl 1230.37112Let \(X\) be a quasiprojective complex variety, let \(\Phi\:X\to X\) be a morphism, and let \(V\) be a subvariety of \(X\). For integers \(i\geq0\), let \(\Phi^i\) denote the \(i\)th iterate \(\Phi\circ\dots\circ\Phi\), and for any \(\alpha\in X(\mathbb C)\) let \(\mathcal O_\Phi(\alpha)\) denote the (forward) \(\Phi\)-orbit \(\{\Phi^i(\alpha):i\geq0\}\) of \(\alpha\). The dynamical Mordell-Lang conjecture is the question of characterizing the set \(\{i\geq0:\Phi^i(\alpha)\in V\}\) for fixed \(\alpha\) and \(V\).

The present paper considers the case in which \(\Phi\) is étale, and shows that under this assumption, if \(\alpha\in X(\mathbb C)\) then the intersection \(\mathcal O_\Phi(\alpha)\cap V(\mathbb C)\) is a union of finitely many orbits of the form \(\mathcal O_{\Phi^N}(\Phi^\ell(\alpha))\) for \(N,\ell\in\mathbb N\).

This theorem gives a new proof of the classical (arithmetic) Mordell-Lang conjecture in the special case of cyclic subgroups.

It also provides a positive answer to Question 11.6 in the paper [Duke Math. J. 126, No. 3, 491–546 (2005; Zbl 1082.14003)] by D. S. Keeler, D. Rogalski and J. T. Stafford. Specifically, it shows that if \(X\) is an integral projective scheme over an algebraically closed field of characteristic zero, and if \(\Phi\) is an automorphism of \(X\), then every Zariski-dense orbit has the property that all of its infinite subsets are also Zariski-dense.

The theorem is proved by \(p\)-adic analytic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. (The methods are made to work over \(\mathbb C\) by first reducing to a subring \(R\) of finite type over \(\mathbb Z\), and then embedding \(R\) into \(\mathbb Z_p\) for some suitable \(p\), so that the geometric properties of the situation are preserved.)

The present paper considers the case in which \(\Phi\) is étale, and shows that under this assumption, if \(\alpha\in X(\mathbb C)\) then the intersection \(\mathcal O_\Phi(\alpha)\cap V(\mathbb C)\) is a union of finitely many orbits of the form \(\mathcal O_{\Phi^N}(\Phi^\ell(\alpha))\) for \(N,\ell\in\mathbb N\).

This theorem gives a new proof of the classical (arithmetic) Mordell-Lang conjecture in the special case of cyclic subgroups.

It also provides a positive answer to Question 11.6 in the paper [Duke Math. J. 126, No. 3, 491–546 (2005; Zbl 1082.14003)] by D. S. Keeler, D. Rogalski and J. T. Stafford. Specifically, it shows that if \(X\) is an integral projective scheme over an algebraically closed field of characteristic zero, and if \(\Phi\) is an automorphism of \(X\), then every Zariski-dense orbit has the property that all of its infinite subsets are also Zariski-dense.

The theorem is proved by \(p\)-adic analytic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. (The methods are made to work over \(\mathbb C\) by first reducing to a subring \(R\) of finite type over \(\mathbb Z\), and then embedding \(R\) into \(\mathbb Z_p\) for some suitable \(p\), so that the geometric properties of the situation are preserved.)

Reviewer: Paul Vojta (Berkeley)