Related papers: The Gross-Kuz'min Connjecture for CM fields

Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

For any positive integer $n$, we show that there exists a real number field $k$ (resp. $k'$) of degree $2^n$ whose $2$-class group is isomorphic $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ such that the Galois group of the maximal…

We prove the $K(\pi,1)$ conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol'd, Pham, and Thom. Our proof is based on…

Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group…

Using Kummer theory for a finite extension K of \Qp(\zeta)(where p is a prime number and \zeta a primitive p-th root of~1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We…

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes…

Kummer's conjecture states that the relative class number of the $p$-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true -- it cannot be true if we assume the truth of two other widely believed…

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in…

Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we…

In 1977 Kervaire and Murthy presented three conjectures regarding $K_0 \mathbb{Z} C_{p^n}$, where $C_{p^n}$ is the cyclic group of order $p^n$ and $p$ is a semi-regular prime. The Mayer-Vietoris exact sequence provides the following short…

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…

In this paper, we describe the higher even $K$-groups of the ring of integers of a number field in terms of class groups of an appropriate extension of the number field in question. This is a natural extension of the previous collective…

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $\rho$ of the absolute Galois group of $K$ is determined…

Kummer's Conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's Conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the…

Let $K/\mathbb{Q}$ be an imaginary quadratic extension, and let $p$ be an odd prime. In this paper, we investigate the growth of Mordell-Weil ranks of CM abelian varieties associated with Hecke characters over $K$ of infinite type $(1, 0)$…

Let $\mathfrak{p}$ be a monic irreducible polynomial in $A:=\mathbb{F}_q[\theta]$, the ring of polynomials in the indeterminate $\theta$ over the finite field $\mathbb{F}_q$, and let $\zeta$ be a root of $\mathfrak{p}$ in an algebraic…

Let p\in\{2,3\}, and let k be an imaginary quadratic field in which p decomposes into two distinct primes \mathfrak{p} and \bar{\mathfrak{p}}. Let k_\infty be the unique Z_p-extension of k which is unramified outside of \mathfrak{p}, and…

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the…