The $S$-unit equation

Abstract: We prove that the $S$-unit equation, $x+y=1$ where $x,y\in\mathcal O_S^\times$, has finitely many solutions by using $p$-adic period mappings. To do so we analyze the monodromy and period mapping for (a small modification of) the Legendre family. Although not logically necessary for the proof of Falting's theorem, many of the key ideas are already present in this special case.

Reference:

$\bullet$ Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 4.

algebraic geometrynumber theory

Audience: advanced learners

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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.

Fall 2025 topic: Weil conjectures.

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.

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