Tag - Projective curves

Xinyi Yuan: A uniform Bogomolov type of theorem for curves over global fields

15th September, 2021

In the recent breakthrough on the uniform Mordell-Lang problem by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform Bogomolov type of theorem for curves over number fields. In this talk, we introduce a refinement and generalization of the uniform Bogomolov conjecture over global fields, as a consequence of bigness of some adelic line bundles in the setting of Arakelov geometry. The treatment is based on the new theory of adelic line bundles of Yuan-Zhang and the admissible pairing over curves of Zhang.

Alexander Carney: Heights and dynamics over arbitrary fields

15th October, 2020

Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and Yuan-Zhang, we show how to define arithmetic intersections and heights relative to any finitely generated field extension 𝐾/π‘˜, and construct canonical heights for polarizable arithmetic dynamical systems 𝑓:𝑋→𝑋. These heights have a corresponding Northcott property when π‘˜ is β„š or π”½π‘ž. When π‘˜ is larger, we show that Northcott for canonical heights is conditional on the non-isotriviality of 𝑓:𝑋→𝑋, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. Additionally, we prove the Hodge Index Theorem for arithmetic intersections relative to 𝐾/π‘˜. Since, when Northcott holds, pre-periodic points are the same as height zero points, this has applications to dynamical systems. By the Lefschetz principle, these results can be applied over any field.