Another look at real quadratic fields of relative class number 1

Debopam Chakraborty; Anupam Saikia

Displaying similar documents to “Another look at real quadratic fields of relative class number 1”

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

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We show that certain quadratic base change $L$ -functions for $Gl (2)$ are non-negative at their center of symmetry.

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

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We describe all sets $A \subseteq_{p}$ which represent the quadratic residues $R \subseteq_{p}$ in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.

Capturing forms in dense subsets of finite fields

Brandon Hanson (2013)

Acta Arithmetica

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An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field $_{q}$ . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size...

Minimal $𝒮$ -universality criteria may vary in size

Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)

Journal de Théorie des Nombres de Bordeaux

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In this note, we give simple examples of sets $𝒮$ of quadratic forms that have minimal $𝒮$ -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.

Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields

Adam Mohamed (2014)

Publications mathématiques de Besançon

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Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫 \subset 𝒪$ be a non-zero ideal and let $p > 5$ be a rational inert prime in $F$ and coprime with $𝔫$ . Let $V$ be an irreducible finite dimensional representation of ${\bar{𝔽}}_{p} [{GL}_{2} (𝔽_{p^{2}})]$ . We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\bar{𝔽}}_{p} \otimes d e t^{e}$ for some $e \geq 0$ ; except possibly in some few cases.

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...

Perfect unary forms over real quadratic fields

Dan Yasaki (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $F = ℚ (\sqrt{d})$ be a real quadratic field with ring of integers $𝒪$ . In this paper we analyze the number $h_{d}$ of ${GL}_{1} (𝒪)$ -orbits of homothety classes of perfect unary forms over $F$ as a function of $d$ . We compute $h_{d}$ exactly for square-free $d \leq 200000$ . By relating perfect forms to continued fractions, we give bounds on $h_{d}$ and address some questions raised by Watanabe, Yano, and Hayashi.

On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous (2016)

Mathematica Bohemica

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We study the capitulation of $2$ -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $𝕜 = ℚ (\sqrt{2 p q}, i)$ , where $i = \sqrt{- 1}$ and $p \equiv - q \equiv 1 (mod 4)$ are different primes. For each of the three quadratic extensions $𝕂 / 𝕜$ inside the absolute genus field $𝕜^{(*)}$ of $𝕜$ , we determine a fundamental system of units and then compute the capitulation kernel of $𝕂 / 𝕜$ . The generators of the groups ${Am}_{s} (𝕜 / F)$ and $Am (𝕜 / F)$ are also determined from which we deduce that $𝕜^{(*)}$ is smaller than the relative genus field ${(𝕜 / ℚ (i))}^{*}$ . Then we prove...