Hilbertian Towers, Frattini
Towers and Serre's O(pen) I(mage) T(heorem)
Consider any cover of normal varieties φ: W → V, defined over a number field K. Suppose p∈W
(an algebraic point) has full Galois closure over V: As in HITcovers.html,
the degree of the Galois closure of K(p)/K(φ(p)) is the same as the degree of the
Galois closure of φ.
The phrase "a dimension u
tower (of varieties) over (a number field) K" (for simplicity take the
varieties affine) we will mean a normal sequence
(*) …→ Xk → Xk-1 →
… → X0
→ Au of finite maps of absolutely
irreducible, dimension u,
varieties with all varieties and maps over K.
Below we will refer to the maps from Xk
to Au by ψk. Now consider these two
properties of the tower (*).
It is an Hilbertian
Tower – with respect to some
fixed integer k0 –
if the following holds:
(*2) If pk0
∈Xk0
has full
Galois closure over Au, then so does any p' ∈Xk
for any k ≥ k0,
and p' lying over pk0.
It is a (geometric) Frattini Tower
– with respect to some
fixed integer k0 –
if the following holds:
(*3) For k ≥ k0,
the geometric monodromy group G(ψk),
of ψk, is a
Frattini extension of G(ψk).
Thm. If (*) is an Hilbertian Tower then it is a (geometric) Frattini
Tower (with respect to the same k0).
Proof: