Let k be a number field, p$\ge$2 a prime and S a set of tame or wild finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k)=Z/p^NZ and if S non-empty is totally ramified. Using analogues of Chevalley's formula (Gras, Proc. Math. Sci. 127(1) (2017)),we give an elementary proof of a stability theorem (Theorem 3.1 for generalized p-class groups X\_n of the layers k\_n$\le$K:let $λ$=max(0, \#S-1-$ρ$) given in Definition 1.1; then\#X\_n = \#X\_0 x p^{$λ$ n} for all n in [0,N], if and only if \#X\_1=\#X\_0 x p^$λ$. This improves the case $λ$ = 0 of Fukuda (1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020),whose techniques are based on Iwasawa's theory or Galois theory of pro-p-groups. We deduce capitulation properties of X\_0 in the tower (e.g. Conjecture 4.1). Finally we apply our principles to the torsion groups T\_n of abelian p-ramification theory. Numerical examples are given.
We consider the Galois group $G_2(K)$ of the maximal unramified $2$-extension of $K$ where $K/\mathbb{Q}$ is cyclic of degree $3$. We also consider the group $G^+_2(K)$ where ramification is allowed at infinity. In the spirit of the Cohen-Lenstra heuristics, we identify certain types of pro-$2$ group as the natural spaces where $G_2(K)$ and $G^+_2(K)$ live when the $2$-class group of $K$ is $2$-generated. While we do not have a theoretical scheme for assigning probabilities, we present data and make some observations and conjectures about the distribution of such groups.
Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The …
In this paper we consider a tower of number fields $\cdots \supseteq K(1) \supseteq K(0) \supseteq K$ arising naturally from a continuous $p$-adic representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/K)$, referred to as a $p$-adic Lie tower over $K$. A recent conjecture of Daqing Wan hypothesizes, for certain $p$-adic Lie towers of curves over $\mathbb{F}_p$, a stable (polynomial) growth formula for the genus. Here we prove the analogous result in characteristic zero, namely: the $p$-adic valuation of the discriminant of the extension $K(i)/K$ is given by a polynomial in $i,p^i$ for $i$ sufficient…
Let p$\ge$2 be a given prime number. We prove, for any number field kappa and any integer e$\ge$1, the p-rank $ε$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers F/kappa built as successive degree p cyclic extensions (without any other Galois conditions) such that F/kappa be of degree p^e, namely: #(Cl\_F[p])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, where D\_F is the absolute value of the discriminant (Theorem 3.6) and, more generally, #(Cl\_F[p^r])<<\_{kappa,p^e,$ε$}($\sqrt$D\_F)^$ε$, for any r$\ge$1 fixed. This Note generalizes the case of the family F\_Q^p (Genus …
Given a fixed prime number \(p\), the multiplet of abelian type invariants of the \(p\)-class groups of all unramified cyclic degree \(p\) extensions of a number field \(K\) is called its IPAD (index-\(p\) abelianization data). These invariants have proved to be a valuable information for determining the Galois group \(G_p^2\) of the second Hilbert \(p\)-class field and the \(p\)-capitulation type \(\varkappa\) of \(K\). For \(p=3\) and a number field \(K\) with elementary \(p\)-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated I…
Let F be a number field and p be a prime. In the Successive Approximation Theorem, we prove that, for each positive integer n, finitely many candidates for the Galois group G(p,n,F) of the n-th stage F(p,n) of the p-class tower F(p,infinity) over F are determined by abelian type invariants of p-class groups Cl(p,E) of unramified extensions E/F with degree [E:F]=$p^{n-1}$. Illustrated by the most extensive numerical results available currently, the transfer kernels ker(T(F,E)) of the p-class extensions T(F,E):Cl(p,F)-->Cl(p,E) from F to unramified cyclic degree-p extensions E/F are shown to be …
Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K), that is the second derived quotient M=G/G". The family tau(1,K) of abelian type invariants of the p-class groups Cl(p,L) of all unramified cyclic extensions L/K of degree p is called the index-p abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different ki…
To extend Iwasawa's classical theorem from ${\mathbb Z}_p$-towers to ${\mathbb Z}_p^d$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a ${\mathbb Z}_p^d$-tower of a global field $K$ ramified at finitely many primes is given by a polynomial in $p^n$ and $n$ of total degree at most $d$ for sufficiently large $n$. This conjecture remains open for $d\geq 2$. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on $p$-adic stability of zeta functions in a $p$-adic Lie tower of funct…
We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.
In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on …
For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G' = C(3^e) x C(3), 2 <= e <= 6, according to the al…
