Related papers: On Gras conjecture for imaginary quadratic fields

For unitary groups associated to a ramified quadratic extension of a $p$-adic field, we define various regular formal moduli spaces of $p$-divisible groups with parahoric levels, characterize exceptional special divisors on them, and…

We prove the Nagell-Lutz theorem for the imaginary quadratic fields of class number one.

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for…

Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid…

We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real…

Let $K/ \mathbb{Q}$ be an imaginary $S_3$-extension, and $p$ a prime number which splits into exactly three primes in $K$. We give a sufficient condition for the validity of Greenberg's generalized conjecture for $K$ and $p$.

Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant…

Based on results obtained in a companion paper [MSRI preprint 1997-002], we construct groups of special $S$--units for function fields of characteristic $p>0$, and show that they satisfy Gras--type Conjectures. We use these results in order…

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not…

Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\…

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…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the…

In the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary…

We prove an analogue of a result by Goldston, Pintz and Yildirim for small gaps between primes that split completely in an abelian number field. We prove both a conditional result assuming the Elliott-Halberstam conjecture, and an…

We prove Rubin's conjecture on the structure of local units in the anticyclotomic $\mathbb{Z}_p$-extension of unramified quadratic extension of $\mathbb{Q}_p$ in $p=3$ case by extending Burungale-Kobayashi-Ota's work.

Let $q$ be a power of a prime number $p$. Let $k=\mathbb{F}_{q}(t)$ be the rational function field with constant field $\mathbb{F}_{q}$. Let $K=k(\alpha)$ be an Artin-Schreier extension of $k$. In this paper, we explicitly describe the…

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of…

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le…

Matom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic…

In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner…