Related papers: On the $\lambda$-stability of p-class groups along…

Let $p$ be a prime and let $F$ be a number field. Consider a Galois extension $K/F$ with Galois group $H\rtimes \Delta$ where $H\cong \mathbb{Z}_p$ or $\mathbb{Z}/p^d\mathbb{Z}$, and $\Delta$ is an arbitrary Galois group. The subfields…

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…

We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion…

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…

We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $\Gamma$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. Our work is inspired by Ozaki's and applies in broader…

Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k,…

The p-class tower $F_p^\infty(k)$ of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group $G:=Gal(F_p^\infty(k)/k)$, can be identified by an explicit…

In a previous paper, the author (together with Matthew Emerton) proved that the completed cohomology groups of SL_N(Z) are stable in fixed degree as N goes to infinity (Z may be replaced by the ring O_F of integers of any number field). In…

Let $k$ be a perfect field of characteristic $p$ and $\Gamma$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any…

Let $k$ be a totally real number field and $p$ a prime. We show that the ``complexity'' of Greenberg's conjecture ($\lambda = \mu = 0$) is of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal…

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 $\epsilon$-conjecture, on the p-class groups Cl\_F, for the family F\_kappa^p^e of towers F/kappa built as successive degree p…

We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $\nu_{L/K}$, we show that…

Let $p$ be an odd prime. For a number field $K$, we let $K_\infty$ be the maximal unramified pro-$p$ extension of $K$; we call the group $\mathrm{Gal}(K_\infty/K)$ the $p$-class tower group of $K$. In a previous work, as a non-abelian…

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…

Let $p$ be a prime. Consider a tower of smooth projective geometrically irreducible curves over $\mathbb F_p$, $\mathscr C:\cdots\rightarrow C_n\rightarrow\cdots\rightarrow C_1\rightarrow C_0=\mathbb P^1$ whose Galois group is isomorphic to…

Let $p$ be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a $\mathbb{Z}_p$-tower. These Galois groups can be considered as non-commutative…

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete…

Our goal in this paper is to investigate four conjectures proposed by Daqing Wan about the stable behavior of a geometric $\mathbb{Z}_p$-tower of curves $X_{\infty}/X$. Let $h_n$ be the class number of the $n$-th layer in $X_{\infty}/X$. It…

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal…

For a fixed prime p, the p-class tower F(p,infinity,K) of a number field K is considered to be known if a pro-p presentation of the Galois group H = Gal( F(p,infinity,K)/K ) is given. In the last few years, it turned out that the Artin…