No abstract

For a semi-abelian variety over a global function field which is isogenous to an isotrivial one, we show that on the product of local points of a subvariety satisfying a minor condition, the topological closure of a finitely generated subgroup of global points cuts out exactly the global points of the subvariety lying in this subgroup. As a corollary, on every non-isotrivial supersingular curve of genus two over a global function field, we conclude that the Brauer-Manin condition cuts out exactly the set of its rational points.

Over a large class of function fields, we show that the solutions of some linear equations in the topological closure of a certain subgroup of the group of units in the function field are exactly the solutions that are already in the subgroup. This result solves some cases of the function field analog of an old conjecture proposed by Skolem.

We propose a function-field analog of Pisot's d-th root conjecture on linear recurrences, and prove it under some "non-triviality" assumption. Besides a recent result of Pasten-Wang on Büchi's d-th power problem, our main tool, which is also developed in this paper, is a function-field analog of an GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such GCD estimate, we also obtain an asymptotic result.

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.