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