Torsion points of elliptic curves over large algebraic extensions of finitely generated fields

The following Theorem is proved:Let K be a finitely generated field over its prime field. Then, for almost all e-tuples (σ)=(σ ₁, ..., σ _e)of elements of the abstract Galois group G(K)of K we have: (a) If e=1, then E _tor(K(σ))is infinite. Morover, there exist infinitely many primes l such that E(K(σ))contains points of order l. (b) If e≧2, then E _tor(K(σ))is finite. (c) If e≧1, then for every prime l, the group E(K(σ))contains only finitely many points of an l-power order. Here K(σ) is the fixed field in the algebraic closure K of K, of σ ₁, ..., σ _e, and "almost all" is meant in the sense of the Haar measure of G(K).