Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek

Displaying similar documents to “Grothendieck and Witt groups in the reduced theory of quadratic forms”

Capturing forms in dense subsets of finite fields

Brandon Hanson (2013)

Acta Arithmetica

Similarity:

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...

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

Similarity:

The relative class number $H_{d} (f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$ , where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$ . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

Positivity of quadratic base change $L$ -functions

Hervé Jacquet, Chen Nan (2001)

Bulletin de la Société Mathématique de France

Similarity:

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

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

Similarity:

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.

Sumsets in quadratic residues

I. D. Shkredov (2014)

Acta Arithmetica

Similarity:

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.

Perfect unary forms over real quadratic fields

Dan Yasaki (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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...

Weighted Erdős-Kac type theorem over quadratic field in short intervals

Xiaoli Liu, Zhishan Yang (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $𝕂$ be a quadratic field over the rational field and $a_{𝕂} (n)$ be the number of nonzero integral ideals with norm $n$ . We establish Erdős-Kac type theorems weighted by $a_{𝕂} {(n)}^{l}$ and $a_{𝕂} {(n^{2})}^{l}$ of quadratic field in short intervals with $l \in ℤ^{+}$ . We also get asymptotic formulae for the average behavior of $a_{𝕂} {(n)}^{l}$ and $a_{𝕂} {(n^{2})}^{l}$ in short intervals.

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

Adam Mohamed (2014)

Publications mathématiques de Besançon

Similarity:

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.