What is... the Manin-Mumford conjecture?
Vortrag von Tobias Bisang
Datum: 01.04.25 Zeit: 16.30 - 18.30 Raum:
This talk is about the Manin-Mumford conjecture, which was proven by M. Raynaud in 1983. The required theory will be presented, including abelian varieties. The first and simplest version of the theorem is as follows: Given a nonzero \(f\in\mathbb{C}[X,Y]\), there exist only finitely many pairs \((q_1,q_2)\in\mathbb{Q}\) with \(f(\mathrm{e}^{2\pi\mathbf{i}q_1},\mathrm{e}^{2\pi\mathbf{i}q_2})=0\), unless there is a very special reason: \(f\) contains a factor of the form \(X^nY^m-\mathrm{e}^{2\pi\mathbf{i}q}\) or \(X^n-Y^m\mathrm{e}^{2\pi\mathbf{i}q}\).