Quadratic Fields Research Articles

Elliptic curves with complex multiplication and abelian division fields

AbstractLet be an imaginary quadratic field, and let be the order in of conductor . Let be an elliptic curve with complex multiplication by , such that is defined by a model over , where . In this article, we classify the values of and the elliptic curves such that (i) the division field is an abelian extension of , and (ii) the ‐division field coincides with the th cyclotomic extension of the base field.

Integral points on a del Pezzo surface over imaginary quadratic fields

Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. In this paper, we generalise the torsor method for integral points from Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{A}}_3$$\\end{document} over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.

Energy change and Landauer’s principle in the interaction between qubit and quantum field theory

We give a general description of the system evolution under the interaction between qubit and quantum field theory up to the second order perturbation, which is also referred to as the simplified model of light-matter interaction. The results are classified into rotating and counter-rotating wave terms, the former corresponding to stimulated absorption and emission, and the latter to Unruh and anti-Unruh effects. We obtain not only the reduced density matrix of the qubit, but also the backreaction obtained by quantum field theory as the environment. The result shows that the energy variation of the quantum field theory is related to trajectory and the initial state of the qubit, the expectation values of the linear and quadratic field operators, and the temporal order product operator. When the qubit is in accelerated motion, the conventional Unruh effect causes the vacuum state to possess a “temperature”, which raises some doubts about the validity of Landauer’s principle. We prove that Landauer’s principle still holds for any state of motion.

On the torsion part in the K-theory of imaginary quadratic fields

We obtain upper bounds for the torsion in the K-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants.

Paving the Way for SQIsign: Toward Efficient Deployment on 32-bit Embedded Devices

The threat of quantum computing has spurred research into post-quantum cryptography. SQIsign, a candidate submitted to the standardization process of the National Institute of Standards and Technology, is emerging as a promising isogeny-based signature scheme. This work aimed to enhance SQIsign’s practical deployment by optimizing its low-level arithmetic operations. Through hierarchical decomposition and performance profiling, we identified the ideal-to-isogeny translation, primarily involving elliptic curve operations, as the main bottleneck. We developed efficient 32-bit finite field arithmetic for elliptic curves, such as basic operations, like addition with carry, subtraction with borrow, and conditional move. We then implemented arithmetic operations in the Montgomery domain, and extended these to quadratic field extensions. Our implementation offers improved compatibility with 32-bit architectures and enables more fine-grained SIMD acceleration. Performance evaluations demonstrated the practicality in low-level operations. Our work has potential in easing the development of SQIsign in practice, making SQIsign more efficient and practical for real-world post-quantum cryptographic applications.

Maximally elastic quadratic fields

Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined assup⁡{sr:π1⋯πr=ρ1⋯ρs, with all πi,ρj irreducible}. We call a quadratic field Kmaximally elastic if the ring of integers of K is a UFD and each element of {1,32,2,52,3,…}∪{∞} appears as an elasticity of infinitely many orders inside K. This corresponds to the orders in K exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that K=Q(2) is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields.

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

Simplified quasi-likelihood analysis for a locally asymptotically quadratic random field

Simplified quasi-likelihood analysis for a locally asymptotically quadratic random field

Characterization and dynamics of certain classes of polynomial vector fields on the torus

In this paper, we classify all polynomial vector fields in R3 of degree up to three such that their flow makes the torusT2={(x,y,z)∈R3:(x2+y2−a2)2+z2−1=0}witha∈(1,∞) invariant. We also classify cubic Kolmogorov vector fields on T2 and prove that they exhibit a rational first integral. We study ‘pseudo-type-n’ vector fields on T2 and show that any such vector field is completely integrable. We prove that the Lie bracket of any two quadratic vector fields on T2 is completely integrable. We explicitly find all cubic vector fields on T2 which achieve the sharp bounds for the number of invariant meridians and parallels. We present necessary and sufficient conditions when invariant meridians and parallels of cubic vector fields on T2 are periodic orbits or limit cycles. We discuss invariant meridians and parallels of pseudo-type-n vector fields as well. Moreover, we characterize the singular points of a class of polynomial vector fields on T2.

Diophantine approximation with prime denominator in quadratic number fields under GRH

Diophantine approximation with prime denominator in quadratic number fields under GRH

An application of Birch–Tate formula to tame kernels of real quadratic number fields

An application of Birch–Tate formula to tame kernels of real quadratic number fields

Non-cyclic torsion of elliptic curves over imaginary quadratic fields of class number 1

Let [Formula: see text] be a non-cyclotomic imaginary quadratic field with class number 1 and [Formula: see text] an elliptic curve with [Formula: see text] In this paper, we determine the torsion groups that can arise as [Formula: see text] where [Formula: see text] is any quadratic extension of [Formula: see text]

Streamlines and neural intelligent scheme for thermal transport to infinite shear rate for ternary hybrid nanofluid subject to homogeneous-heterogeneous reactions

SignificanceThe CuO, Al2O3, and TiO2 nanoparticles find extensive applications in advanced chemical reaction-based thermal transport and quadratic convective nanofluids due to their exceptional thermal properties and chemical stability. Increased thermal conductivity of these nanoparticles used to enhance heat transfer in various chemical reactors, resistance to chemical degradation and photocatalytic reactors and solar energy applications. MotiveThis study brings the investigation about quadratic convection-based thermal transport to infinite shear rate for magnetized ternary radiative cross nanofluid with homogeneous-heterogeneous chemical reactions. Water is taken as base fluid and three nanoparticles are Copper oxide (CuO), aluminium oxide (Al2O3), and titanium dioxide (TiO2). Heat transport analysis is made through quadratic convection, magnetic field and thermal radiation. Concentration of nanofluid is scrutinized though homogeneous-heterogeneous chemical reactions. Methodology: Physical problem with assumptions generates the system of partial differential equations (PDEs) and these PDEs are transformed into ordinary differential equations (ODEs) through similarity variables. Furthermore, a unique combination of Bvp4c and Levenberg Marquardt neural network (LM-NN) schemes is utilized to fetch the numerical solutions. Bvp4c is utilized to solve the governing equations, while LM-NN serves to enhance predictive capabilities and capture intricate nonlinear relationships. FindingsMagnetic environment, chemical process, radiations effects and volumetric fraction of nanoparticles make better heat transfer efficiency and control.

Class group and factorization in orders of a PID

In this paper, we study properties of factorization in orders of a PID via the computation of algebraic invariants that measure the failure of unique factorization. The focus is on the numerical semigroup rings over a finite field and the orders of imaginary quadratic fields with class number 1. We also give a complete description of the class group structure of those rings.

Scalarization of Kerr–Newman black holes in the Einstein–Chern–Simons-scalar theory

We investigate the tachyonic instability of Kerr–Newman (KN) black hole with a rotation parameter a in the Einstein–Chern–Simons-scalar theory coupled with a quadratic massive scalar field. This instability analysis corresponds to exploring the onset of spontaneous scalarization for KN black holes. First, we find no a-bound for α<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha <0$$\\end{document} case by considering (1 + 1)-dimensional analytical method. A direct numerical method is adopted to explore (2 + 1)-dimensional time evolution of a massive scalar perturbation with positive and negative α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to obtain threshold curves numerically. We obtain threshold curves αth(a)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _{\ extrm{th}}(a)$$\\end{document} of tachyonic instability for positive α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} without any a-bounds. We expect to find the same threshold curves αth(a)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha _{\ extrm{th}}(a)$$\\end{document} of tachyonic instability for negative α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} without any a-bound because its linearized scalar theory is invariant under the transformation of α→-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow -\\alpha $$\\end{document} and θ→-θ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\rightarrow -\ heta .$$\\end{document} In addition, it is found that the scalar mass term suppresses tachyonic instability of KN black holes, while the black hole charge promotes the instability (disregarding the contribution of the charge to the existence conditions M2-Q2≥a2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^2-Q^2\\ge a^2)$$\\end{document}.

Two-photon charging of a quantum battery with a Gaussian pulse envelope

Quantum energy science is rapidly emerging as a domain interested in the generation, transfer and storage of energy at the quantum level. In particular, quantum batteries have the scope to exploit the wonders of quantum mechanics in order to boost their performance as compared to their electrochemical equivalents. Here we show how an exponential enhancement in stored energy can be achieved with a quantum battery thanks to a two-photon charging protocol. We consider theoretically a quantum battery modelled as a quantum harmonic oscillator, which when driven by a quadratic field (manifested by a Gaussian pulse envelope) gives rise to squeezing of the battery. This quantum squeezing ensures that the population of the battery is driven exponentially up its bosonic energy ladder. Our results demonstrate a plausible mechanism for quickly storing a vast amount of energy in a quantum object defined by continuous variables, which may be explored experimentally in a variety of quantum optical platforms.

ℓ-adic digits and class number of imaginary quadratic fields

Motivated by the work of [K. Girstmair, A “popular” class number formula, Amer. Math. Monthly 101(10) (1994) 997–1001; K. Girstmair, The digits of [Formula: see text] in connection with class number factors, Acta Arith. 67(4) (1994) 381–386] and [M. R. Murty and R. Thangadurai, The class number of [Formula: see text] and digits of [Formula: see text], Proc. Amer. Math. Soc. 139(4) (2010) 1277–1289], we study the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] whenever [Formula: see text] is a product of two distinct primes or a prime power. More explicitly, if [Formula: see text] is an integer such that [Formula: see text] and suppose that [Formula: see text] is the [Formula: see text]-adic expansion of [Formula: see text] then we establish the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] in terms of [Formula: see text] and the “trace” of generalized Bernoulli numbers [Formula: see text] where [Formula: see text]’s are odd Dirichlet characters modulo [Formula: see text] As a consequence of these results, we recover two well-known results of Gauss and Heilbronn (see Theorems 1.6 and 1.7).

Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets

We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map [Formula: see text]. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map [Formula: see text] and of its inverse [Formula: see text]. We show that the map [Formula: see text] is the product of two involutions. The nature of the fixed points of [Formula: see text] is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping [Formula: see text] is integrable for [Formula: see text] and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for [Formula: see text].

Imaginary Quadratic Fields With ℓ-Torsion-Free Class Groups and Specified Split Primes

Abstract Given an odd prime $\ell $ and finite set of odd primes $S_{+}$, we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell $ and which splits at every prime in $S_{+}$. Notably, we do not require that $p \not \equiv -1 \,\;(\mathrm{mod}\, \ell )$ for any of the split primes $p$ that we impose. Our theorem is in the spirit of a result by Wiles, but we introduce a new method. It relies on a significant improvement of our earlier work on the classification of non-holomorphic Ramanujan-type congruences for Hurwitz class numbers.

$$\mathbb {Z}_2$$-extension of real quadratic fields with $$\mathbb {Z}/2\mathbb {Z}$$ as 2-class group at each layer

$$\mathbb {Z}_2$$-extension of real quadratic fields with $$\mathbb {Z}/2\mathbb {Z}$$ as 2-class group at each layer