Title: Finiteness results in number theory (Part I)

Speaker:　Ariyan Javanpeykar

Institute: University of Mainz, Germany

Time: May 5th, Friday 16:00-17:30

Place:1518

Abstract: The set of rational points on an elliptic curve over a number field is finitely generated by a theorem of Mordell and Weil. In these lectures, we will first explain the main ideas of the proof of the latter theorem. A part of the proof of the Mordell-Weil theorem fits in well with a philosophy of Shafarevich. Namely, the set of "objects of fixed type" over a fixed number ring should be finite. I will give precise examples illustrating Shafarevich's philosophy with abelian varieties, hyperbolic curves, and K3 surfaces and explain the relation to the Lang-Vojta conjecture. In the course of lectures, the proof of Faltings' finiteness theorem will be also mentioned.