Height bounds for nondegenerate varieties

Abstract: We will prove the Silverman-Tate theorem in Appendix 5 of [DGH], which upper-bounds the difference between the Neron-Tate height and the Weil height of a point $P$ in an abelian scheme $\pi: \mathcal{A}\to S$ in terms of the height of the point $\pi(P)$ in the base scheme. Then, we'll apply this result, together with last week's Proposition 4.1 of [DGH], to prove Theorem 1.6 in [DGH], which gives a lower bound on the Neron-Tate height of $P$ in a nondegenerate subvariety $X$ of $\mathcal{A}\to S$ in terms of the height of $\pi(P)$. For this application, we follow Section 5 of [DGH].

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.