This course focuses on the dynamical Mordell-Lang Conjecture. We first describe the Mordell-Lang Conjecture (Faltings' theorem) for abelian varieties. We then describe the dynamical analog, which predicts an algebraic structure for the intersection of an orbit with a subvariety. Next, we will discuss the one-dimensional case, which is better-known as the Skolem-Mahler-Lech theorem describing the periodicity of zeroes in linear recurrence sequences. We then discuss a higher-dimensional counterexample to the dynamical Mordell-Lang conjecture, given by Scanlon and Yasufuku. We end with some positive results in higher-dimensions: Bell-Ghioca-Tucker's result for etale morphisms, and Xie's result for polynomial maps on affine plane which heavily uses Favre-Jonsson's valuative dynamical compactification.
We are still searching for the correct formulation of the dynamical Mordell-Lang conjecture, so it is fruitful to prove new cases as well as to find counterexamples. The compactification method is only available up to dimension two so far, so a more general theory is necessary. By introducing a research topic which allows for both theoretical and example-oriented approaches, the hope is that more students become interested in the dynamical Mordell-Lang problem and related fields.
・To understand the statement of the Mordell-Lang conjecture for abelian varieties
・To learn about p-adic analytic functions and their application to the Skolem-Mahler-Lech Theorem on linear recurrences
・To learn about compactification using valuations, and their application to the Mordell-Lang problem for affine planes
arithmetic dynamics, abelian varieties, Mordell-Lang conjecture, linear recurrences, p-adic analytic functions, compactifications using valuations
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||The following topics will be covered in this order: -- the Mordell-Lang Conjecture (Faltings' theorem) for abelian varieties -- The first formulation of the dynamical Mordell-Lang Conjecture -- p-adic analytic functions -- 1-dimensional case: Skolem-Mahler-Lech's Theorem for linear recurrences -- Higher-dimensional counterexamples: examples of Scanlon-Yasufuku -- The second formulation of the dynamical Mordell-Lang Conjecture -- Etale Morphisms: Bell-Ghioca-Tucker's Theorem -- Favre-Jonsson's dynamical compactification using valuations on affine plane -- Xie's result for (f(x,y), g(x,y)) using dynamical compactification||Details will be provided during each class session|
Bell, Ghioca, Tucker 「The Dynamical Mordell-Lang Conjecture」 AMS (2016)
Favre, Jonsson 「The Valuative Tree」 Springer LNM 1853 (2004)