Seminar

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.