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

Let $K$ be the $N$th layer in the cyclotomic $\widehat{\mathbb{Z}}$-extension $\widehat{\mathbb{Q}}$. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington,\,$\ldots$) analyse the…

Let $p$ be an odd rational prime and consider the cyclotomic number field $K = \mathbb{Q}(\zeta_{p})$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen--Franks group…

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime where $E$ has ordinary reduction and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. We give sufficient…

Let $N=K(\sqrt[3]{D})$ be a cubic Kummer extension of the cyclotomic field $K=\mathbb{Q}(\zeta_3)$, containing a primitive cube root of unity $\zeta_3$, with cube free integer radicand $D>1$. Denote by $f$ the conductor of the abelian…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

Let $\ell$ be prime, and $K$ be a number field containing the $\ell$-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of $\mathbb Z/\ell\mathbb Z$ extensions of $K$…

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…

Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim H^1(G_{K},\mathbb{F}_p)<\infty$ and $\dim H^2(G_{K},\mathbb{F}_p)=1$, we show that L.~Positselski's…

For an algebraic number field K such that prime l splits completely in K we define a regulator R(K) that characterize the subgroup of universal norms from the cyclotomic extension of K in the completed group of S-units of K, where S…

For a prime $p\equiv 1 \,(\bmod{4})$, let \[ \varepsilon = \frac{1}{2}\left( t + u\sqrt{p}\right) \] be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In 1951, N. Ankeny, E. Artin, and S. Chowla asked whether $p$…

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…

The $p$-adic Littlewood Conjecture due to De Mathan and Teuli\'e asserts that for any prime number $p$ and any real number $\alpha$, the equation $$\inf_{|m|\ge 1} |m|\cdot |m|_p\cdot |\langle m\alpha \rangle|\, =\, 0 $$ holds. Here, $|m|$…

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…

Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they…

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that…

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 L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of…

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH).…

Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero and…

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 $(1, 0)$…