Going Beyond Current Limits:

We just invented the concept of Iterated IPADs.
This is a revolutionary new method
of filtering possible ptower groups
by computing Invariants of Second Order:

Iterated IPAD Sweeps 2Tower Groups

The New Method:

*************************************************************************
For each unramified cyclic extension LK (of 1st order)
of degree p over a given base field K, we additionaly determine
its unramified cyclic degree p extensions NL of 2nd order.
Consequently, we end up getting an iterated or extended IPAD
(indexp abelianization data) τ_{p}^{2}(K),
that is a family of IPADs [τ(K);τ(L_{1});...;τ(L_{n})] with suitable n.
(We avoid the terminology "extended TTT", since we do not know yet
whether these new abelianizations are associated to transfers.)
*************************************************************************
We briefly explain our ideas by three examples,
the first two in connection with threestage towers of 2class fields.


Example 1. p = 2, K = Q((1780)^{1/2}) with Cl_{2}(K) ~ (2,4) has n = 3 and
τ_{2}^{2}(K) = [τ(K);τ(L_{1});...;τ(L_{3})] =
[(2,2,2),(4,4),(2,8);
(2,2,4)^{6},(4,4);
(2,2,4)^{3};
(2,2,4),(2,16)^{2}],
as mentioned and used in
M. R. Bush's 2003 paper
.
Of course, the extension K((13+4*5^{1/2})^{1/2})
cannot be cyclic or abelian over K. It must be nonGalois!


Example 2. p = 2, K = Q((2067)^{1/2}) with Cl_{2}(K) ~ (2,4) has n = 3 and
τ_{2}^{2}(K) = [τ(K);τ(L_{1});...;τ(L_{3})] =
[(2,2,2),(4,4),(2,8);
(2,2,2,2)^{3},(2,2,8)^{2},(2,2,4),(4,4);
(2,2,8)^{2},(2,2,4);
(2,2,4),(4,8)^{2}].


And now some hopeful expectations,
shedding light on the infinite family by
L. Bartholdi and M. R. Bush, 2007
:
Example 3. p = 3, K = Q((3896)^{1/2}) with Cl_{3}(K) ~ (3,3) has n = 4 and
τ_{3}^{2}(K) = [τ(K);τ(L_{1});...;τ(L_{4})] =
[(3,3,3),(3,9),(3,3,3),(3,3,3);
(3,3,9)^{4},(3,3)^{9};
(3,3,9),(3,27)^{3};
(3,3,9)^{4},(9,9)^{9};
(3,3,9)^{4},(9,9)^{9}].
For these fields of absolute degree 18,
Magma has to struggle considerably, but it finally succeeds.
Similar results were obtained for the discriminants
d = –6583,23428,25447,27355,27991,36276,37219,37540.


We should now give an exact definition of the iterated IPAD
and emphasize that
its interpretation via transfers seems to be problematic.
We cannot see how to express the components of second order
by means of any higher pclass group Gal(F_{p}^{s}(K)K), s > 1,
of a given base field K.

Definition.
Let p be a prime,
r > 1 an integer,
KQ a number field with pclass group Cl_{p}(K) of rank r,
m = (p^{r}1)/(p1), and
(L_{1},...,L_{m}) the family of unramified cyclic extensions of K
of relative degree p and absolute degree p*[K:Q].
For each i in {1,...,m}, let
r_{i} be the pclass rank of L_{i},
n_{i} = (p^{r_{i}}1)/(p1), and
(N_{i,j})_{j in {1,...,n_{i}}}
the family of unramified cyclic extensions of L_{i}
of relative degree p and absolute degree p^{2}*[K:Q].
The iterated IPAD τ_{p}^{2}(K) of K consists of
a single component of first order,
[Cl_{p}(L_{i}),...,Cl_{p}(L_{m})],
and m components of second order,
[Cl_{p}(N_{i,j})]_{j in {1,...,n_{i}}} for i in {1,...,m}.


Remark 1.
For p = 2 and a base field K of type (2,4),
the iterated IPAD contains the second layer of the TTT τ(K) of K,
since m = 3 and there exists i in {1,...,3},
such that L_{i} is common subfield of the 3
unramified extensions of relative degree 4 of K.


Remark 2.
For p = 3 and a base field K of type (3,3),
each component of second order of τ_{3}^{2}(K) includes
the abelian type invariants of the Hilbert 3class field of K,
since m = 4 and
for each i in {1,...,4} there exists j in {1,...,n_{i}},
such that N_{i,j} = F_{3}^{1}(K).


While there is no difficulty to form composita of
class extension homomorphisms:
J(L_{i}K) : Cl_{p}(K) → Cl_{p}(L_{i}) and
J(N_{i,j}L_{i}) : Cl_{p}(L_{i}) → Cl_{p}(N_{i,j}) yield together
J(N_{i,j}L_{i}) * J(L_{i}K) =
= J(N_{i,j}K) : Cl_{p}(K) → Cl_{p}(N_{i,j}),
the corresponding transfers from the pgroups
G = Gal(F_{p}^{s}(K)K), s > 1, resp.
G_{i} = Gal(F_{p}^{t}(L_{i})L_{i}), t > 1,
to their subgroups
H_{i} = Gal(F_{p}^{s}(K)L_{i}), resp.
H_{i,j} = Gal(F_{p}^{t}(L_{i})N_{i,j}),
T(G,H_{i}) : G/G' → H_{i}/H_{i}' and
T(G_{i},H_{i,j}) : G_{i}/G_{i}' → H_{i,j}/H_{i,j}'
cannot be concatenated immediately.
An isomorphism
f : H_{i}/H_{i}' ~ Gal(F_{p}^{1}(L_{i})L_{i}) ~ G_{i}/G_{i}'
must be inserted to obtain a formal compositum
T(G_i,H_{i,j}) * f * T(G,H_{i}) =
T(G,H_{i,j}) : G/G' → H_{i,j}/H_{i,j}'
and we have doubts whether this formal compositum
can be viewed as a transfer,
since H_{i,j} is not a subgroup of G,
and whether it can be associated to J(N_{i,j}K),
in general with nonabelian extension N_{i,j}K.




Some Caveats:

It seems that this nonabelian problem
must be tackled from the
viewpoint of projective limits,
namely the ptower group
Gal(F_{p}^{∞}(K)K)
of the maximal unramified
prop extension of K.

Finally it should be remarked that
we are trying to identify
G = Gal(F_{p}^{s}(K)K) via τ_{p}^{2}(K),
but G_{i} = Gal(F_{p}^{t}(L_{i})L_{i}) is
(and will probably remain)
completely unknown.
