MSRI - Basic Notions Seminar: "The geometric Mordell conjecture"

Time:	2:00PM - 3:00PM
Location:	Simons Auditorium
Speakers:	Sandor Kovacs, Sandor Kovacs
Abstract:	Let X be a smooth curve of genus at least 2 over a field F. The Mordell conjecture roughly states that under some clearly necessary conditions X has finitely many rational points over F. If F is a number field then one obtains the arithmetic version and if F is the function field of a curve, then the geometric version. In this talk I will discuss how this theorem follows from another famous conjecture, that of Shafarevich and how one might go about proving the latter in the geometric case. The reduction step works in all characteristic and is due to Parshin. The geometric Mordell conjecture was first proved by Manin and the geometric Shafarevich conjecture (and hence Mordell via Parshin's reduction step) by Parshin in a special case and by Arakelov in general. The arithmetic Shafarevich conjecture was proved by Faltings on his way of proving the (arithmetic) Mordell conjecture. Time permitting I will discuss various higher dimensional generalizations of the Shafarevich conjecture.