We will speak about the following theorem:
Every non-zero abelian variety over an ample field of
characteristic zero has infinite rank.
The proof makes use of a deep theorem of Faltings, Vojta,
Raynaud and Hindry (the ex Mordell-Lang conjecture).
We give several applications of this theorem.
One such application is the construction of examples of non-ample
infinite-algebraic extensions of Q.
To achieve this we view the above theorem as
a sufficient condition for non-ampleness
and combine it with results in Iwasawa theory.