Related papers: On Hilbert genus fields of imaginary cyclic quarti…

Suppose $F=\mathbb{Q}(\sqrt{-p_1\dotsm p_t})$ is an imaginary quadratic number field with distinct primes $p_1,\dots,p_{t}$, where $p_i\equiv 1\pmod{4}$ ($i=1,\dots,t-1$) and $p_t\equiv 3\pmod{4}$. We express the possible values of the…

Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$…

Let $d$ be an odd square-free integer and $\zeta_8$ a primitive $8$-th root of unity. The purpose of this paper is to investigate the rank of the $2$-class group of the fields $L_d=\mathbb{Q}(\zeta_8,\sqrt{d})$.

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…

Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an…

The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue…

Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for $p$ and $k$, which states that the non-trivial Iwasawa module of the…

This is the second in a series of two papers presenting a solution to Hilbert's 12th problem for real quadratic function fields in positive characteristic, in the sense of proving an analog of the Theorem of Weber-Fueter. We also offer a…

In this paper we extend methods of Rubin to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p which divide the number of roots of unity in k.

We suspect that the ``genus part'' of the class number of a number field K may be an obstruction for an ``easy proof'' of the classical p-rank epsilon-conjecture for p-class groups and, a fortiori, for a proof of the ``strong…

For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of…

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_d$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_d \gg (\log d)^2$ almost always.…

Let $\mathds{k}$ be a real quadratic number field. Denote by $\mathrm{Cl}_2(\mathds{k})$ its $2$-class group and by $\mathds{k}_2^{(1)}$ (resp. $\mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field. The aim of this paper is…

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac…

In this paper we obtain an asymptotic formula for the number of $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\bZ$ having bounded discriminant $\Delta = 1-4p$, with $p$ a prime. We…

We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an)^2+4a$ for positive odd integers $a$ and $n$.

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 $d \in \{-4, -8, 8\}$. We study the $8$-part of the narrow class group in the thin families of quadratic number fields of the form $\mathbb{Q}(\sqrt{dpq})$, where $p\equiv q \equiv 1\bmod 4$ are prime numbers, and we prove new lower…

A curious identity of Bunyakovsky (1882), made more widely known by P\'olya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv…

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion…