Erratum “On the computation of quadratic $2$-class groups”

W. Bosma; P. Stevenhagen

Displaying similar documents to “Erratum “On the computation of quadratic $2$ -class groups””

Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek (1980)

Annales Polonici Mathematici

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Abstract. Let F be a formally real field. Denote by G(F) and $G_{t} (F)$ the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group $G (F) / G_{t} (F)$ . This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for $G (F) / G_{t} (F)$ , which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the...

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.

High-order phase transitions in the quadratic family

Daniel Coronel, Juan Rivera-Letelier (2015)

Journal of the European Mathematical Society

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We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as $x \mapsto \exp (- x^{- 2})$ near $x = 0$ , before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

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...

The expansion of $\prod_{k = 1}^{\infty} (1 - q^{a k}) (1 - q^{b k})$

Zhi-Hong Sun (2008)

Acta Arithmetica

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Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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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...

The sum of divisors of a quadratic form

Lilu Zhao (2014)

Acta Arithmetica

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We study the sum τ of divisors of the quadratic form m₁² + m₂² + m₃². Let $S ₃ (X) = \sum_{1 \leq m ₁, m ₂, m ₃ \leq X} τ (m ₁ ² + m ₂ ² + m ₃ ²)$ . We obtain the asymptotic formula S₃(X) = C₁X³logX + C₂X³ + O(X²log⁷X), where C₁,C₂ are two constants. This improves upon the error term $O_{ε} (X^{8 / 3 + ε})$ obtained by Guo and Zhai (2012).

On the number of representations of a positive integer by certain quadratic forms

Ernest X. W. Xia (2014)

Colloquium Mathematicae

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For natural numbers a,b and positive integer n, let R(a,b;n) denote the number of representations of n in the form $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j})$ . Lomadze discovered a formula for R(6,0;n). Explicit formulas for R(1,5;n), R(2,4;n), R(3,3;n), R(4,2;n) and R(5,1;n) are determined in this paper by using the (p;k)-parametrization of theta functions due to Alaca, Alaca and Williams.

Representation numbers of five sextenary quadratic forms

Ernest X. W. Xia, Olivia X. M. Yao, A. F. Y. Zhao (2015)

Colloquium Mathematicae

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For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j}) + 4 \sum_{k = 1}^{c} (r ²_{k} + r_{k} s_{k} + s ²_{k})$ . Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

Binary quadratic forms and Eichler orders

Montserrat Alsina (2005)

Journal de Théorie des Nombres de Bordeaux

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For any Eichler order $𝒪 (D, N)$ of level $N$ in an indefinite quaternion algebra of discriminant $D$ there is a Fuchsian group $Γ (D, N) \subseteq SL (2, ℝ)$ and a Shimura curve $X (D, N)$ . We associate to $𝒪 (D, N)$ a set $ℋ (𝒪 (D, N))$ of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to $Γ (D, N)$ , for primitive forms contained in $ℋ (𝒪 (D, N))$ . In particular, the classification theory of primitive integral binary quadratic forms by $SL (2, ℤ)$ is recovered. Explicit fundamental domains for $Γ (D, N)$ allow the characterization...

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.

Tameness criterion for posets with zero-relations and three-partite subamalgams of tiled orders

Stanisław Kasjan (2002)

Colloquium Mathematicae

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A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if $Λ^{•}$ is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form $q_{Λ^{•}}$ associated with $Λ^{•}$ in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28]...

High Frequency limit of the Helmholtz Equations

Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame (1999-2000)

Séminaire Équations aux dérivées partielles

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We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not...

Displaying similar documents to “Erratum “On the computation of quadratic 2 -class groups””

Displaying similar documents to “Erratum “On the computation of quadratic $2$ -class groups””