Quadratic Fields Research Articles

A weak form of Greenberg's generalized conjecture for imaginary quadratic fields

Let p be an odd prime number and k an imaginary quadratic field in which p splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for p and k, which states that the non-trivial Iwasawa module of the maximal multiple Zp-extension field over k has a non-trivial pseudo-null submodule. We prove this conjecture for p and k under the assumption that the Iwasawa λ-invariants vanish for the Zp-extensions over k in which one of the prime ideals of k lying above p do not ramify and that the characteristic ideal of the Iwasawa module associated to the cyclotomic Zp-extension over k has a square-free generator.

Bounds for 2-Selmer ranks in terms of seminarrow class groups

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]/{(F(x))}$, which we call the \textit{semi-narrow class group} of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime. As an application, when $K$ is a real quadratic field, $E/K$ is semistable and the discriminant of $F$ is totally negative, then we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.

Distribution of toric periods of modular forms on definite quaternion algebras

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on $\mathcal{O}$, we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic $L$-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients $\{a(n)\}_n$ of weight 3/2 modular forms, where $n$ ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.

Linear quadratic mean field social control with common noise: A directly decoupling method

This paper is concerned with mean field linear quadratic social control with common noise, where the weight matrices of individual costs are indefinite We first obtain a set of forward–backward stochastic differential equations (FBSDEs) from variational analysis, and then construct a centralized feedback representation by decoupling the FBSDEs. By using solutions of two Riccati equations, we design a set of decentralized control laws, which is further shown to be asymptotically social optimal. The necessary and sufficient conditions are given for uniform stabilization of the systems by exploiting the relation between population state average and aggregate effect. An explicit expression of the optimal social cost is given in terms of two Riccati equations. Besides, the decentralized optimal solution is provided for mean field social control problem with finite agents.

Improved single snapshot beamforming in a noisy environment using the cubic autoproduct

The presence of random noise is a challenge for remote sensing tasks as success typically diminishes with decreasing signal-to-noise ratio (SNR). To improve performance in noisy environments, signal processing techniques commonly implement snapshot averaging, requiring multiple signal samples. This presentation investigates the improvement offered by the cubic autoproduct for low SNR beamforming with a single snapshot. Prior work has demonstrated the utility of a quadratic product of complex field amplitudes within the signal bandwidth for beamforming and matched field applications. This quadratic field product, known as the frequency-difference autoproduct, is a synthetic estimate of an acoustic field at the difference frequency of the two constituent fields. That formulation is extended here to a cubic product of three complex field amplitudes within the signal bandwidth, termed the cubic autoproduct, capable of mimicking field content at frequencies below, above, and within the signal bandwidth. Important features and the mathematical description of the cubic autoproduct are reviewed before discussing the beamforming approach. Single snapshot beamforming results from experimental data, acquired in a noisy underwater environment, are directly compared between the cubic frequency-difference autoproduct and the acoustic field. [Work supported by ONR and by the US DoD through an NDSEG Fellowship.]

Linear Forms in Polylogarithms

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $\Phi_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic function. Let $\alpha_1, \cdots, \alpha_m$ be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with $0<|\alpha_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion for the linear independence, over an algebraic number field containing $\mathbb{Q}(\alpha_1, \cdots, \alpha_m)$, of all the $rm+1$ numbers : $\Phi_1(x,\alpha_1)$, $\Phi_2(x,\alpha_1), $ $\cdots , \Phi_r(x,\alpha_1)$, $\Phi_1(x,\alpha_2)$, $\Phi_2(x,\alpha_2), $ $\cdots , \Phi_r(x,\alpha_2), \cdots, \cdots, \Phi_1(x,\alpha_m)$, $\Phi_2(x,\alpha_m)$, $\cdots , \Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [10], together with a proof in detail on the non-vanishing Wronskian of Hermite type.

Bias of root numbers for Hilbert newforms of cubic level

We give a general formula of the bias of root numbers for Hilbert modular newforms of cubic level. Explicit calculation is given when the base field is Q,Q(2),Q(5) and the level is the cube of certain rational integers. This complements a previous result of the second author and extends the bias phenomenon to the number fields. Our method is based on Jacquet-Zagier's trace formula, and the explicit calculation works generally for all real quadratic fields of narrow class number one and for rational cubic levels.

Small Amplitude Periodic Orbits in Three-Dimensional Quadratic Vector Fields with a Zero-Hopf Singularity

We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} of revolutions about a rotation axis and then returns to the initial point closing the orbit. When the parameters of the family are restricted to certain explicitly computable open semi-algebraic sets Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}, we characterize those parameters that give rise to the appearance of local two-dimensional periodic invariant manifolds through the singularity. Also we use a Bautin-type analysis to study the maximum number of small-amplitude ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-limit cycles that can be made to bifurcate from the equilibrium when the parameters of the family are restricted to Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}. We obtain global upper bounds on the number of bifurcated ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-limit cycles.

Infinite families of class groups of quadratic fields with 3-rank at least one: quantitative bounds

We improve an effective lower bound on the number of imaginary quadratic fields whose absolute discriminants are less than or equal to [Formula: see text] and whose ideal class groups have 3-rank at least one, which is [Formula: see text]. We also obtain a better bound on the number of imaginary quadratic fields with 3-rank at least two, which is [Formula: see text]; the best-known lower bound so far is [Formula: see text]. For finding these effective lower bounds, we use the Scholz criteria and the parametric families of quadratic fields [Formula: see text] and [Formula: see text] (defined as follows) with escalatory case. We find new infinite families of quadratic fields [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are integers subject to certain conditions for [Formula: see text]. More specifically, we find a complete criterion for the 3-rank difference between [Formula: see text] and its associated quadratic field [Formula: see text] to be one; this is the escalatory case. We also obtain a sufficient condition for the family [Formula: see text] and its associated family [Formula: see text] to have escalatory case. We illustrate some selective implementation results on the 3-class group ranks of [Formula: see text] and [Formula: see text] for [Formula: see text].

Class number one problem for a family of real quadratic fields

Class number one problem for a family of real quadratic fields

The mean number of 3-torsion elements in ray class groups of quadratic fields

We determine the average number of 3-torsion elements in the ray class groups of a fixed (integral) conductor c of quadratic fields ordered by their absolute discriminant, generalizing Davenport and Heilbronn’s theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor c is taken to be a squarefree integer having very few prime factors none of which are congruent to 1 mod 3. Additionally, we compute the second main term for the number of 3-torsion elements in ray class groups with a fixed conductor of quadratic fields ordered by their discriminant.

Congruences of Hurwitz class numbers on square classes

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadratic fields and Ramanujan-type congruences for Hurwitz class numbers.

Unramified extensions of quadratic number fields with certain perfect Galois groups

In this paper, we provide infinite families of quadratic number fields with everywhere unramified Galois extensions of Galois group [Formula: see text] and [Formula: see text], respectively. To my knowledge, these are the first instances of infinitely many such realizations for perfect groups which are not generated by involutions, a property which makes them difficult to approach for the problem in question and leads to somewhat delicate local–global problems in inverse Galois theory.

On some symmetries of the base n expansion of 1∕m : the class number connection

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Furthermore, we obtain corollaries connecting the $ 3 $ rank of the class number of the real quadratic field $ \mathbb{Q}(\sqrt{m}) $ to the $ 3 $ divisibility of the number of certain quadratic residues modulo $ m $. Secondly, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain distribution results of $ m $ modulo any prime $ p $ that properly divides $ n+1 $. Our result implies that the primes $ m $ for which $ n $ is a primitive root belong to one of two equivalence classes modulo any prime $ p $ as above.

Ro-vibrational levels and their (e-f) splitting of acetylene molecule calculated from new potential energy surfaces

Ro-vibrational energy levels of acetylene are reported using variational nuclear motion calculations from new ab initio and empirically optimized full six-dimensional potential energy surfaces in the ground electronic state of the acetylene molecule. The calculations account for the triple, quadruple and quintuple excitations as well as relativistic and diagonal Born-Oppenheimer corrections. Variational nuclear motion calculations were performed using the exact kinetic energy operator in orthogonal coordinates. The convergence of energy levels calculations versus the size of the vibrational basis set functions was verified. Our best ab initio potential energy surface that includes the above-mentioned contributions provides the RMS (obs.-calc.) errors of 0.95 cm−1 for five fundamental energy levels. The largest contribution to the RMS error is caused primarily by a significant deviation of the ν4 fundamental frequency. Experimental values of 120 vibrational band origins were used to empirically adjust few lower-order parameters of the potential energy surface. The average error drops down to 0.45 cm−1 or 0.25 cm−1 for empirically optimized potential energy function with two or seven adjusted parameters corresponding to quadratic force field terms and one third order term. The splitting of (e – f) rovibrational doublets and their J dependence were calculated with high accuracy due to the full account of Coriolis interactions. Computed (e – f) splittings allow one to check the correctness of the assignment of empirical energy levels. The estimation of the accuracy for the calculated vibrational levels in an extended range up to 9500 cm−1 shows that the set of ab initio vibrational levels can be used for future assignments of empirically not observed ro-vibrational energy levels. The comparison of the calculated and experimental ro-vibrational energy levels of the C2D2 and 13C2H2 isotopologues is also reported.

Galois representations attached to elliptic curves with complex multiplication

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $\rho_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\to \operatorname{GL}(2,\mathbb{Z}_p)$ be the Galois representation associated to the Galois action on the Tate module $T_p(E)$. The goal is then to describe, explicitly, the groups of $\operatorname{GL}(2,\mathbb{Z}_p)$ that can occur as images of $\rho_{E,p^\infty}$, up to conjugation, for an arbitrary order $\mathcal{O}_{K,f}$.

On the Problem of Describing Elements of Elliptic Fields with a Periodic Expansion into a Continued Fraction over Quadratic Fields

On the Problem of Describing Elements of Elliptic Fields with a Periodic Expansion into a Continued Fraction over Quadratic Fields

Quadratic rational maps with integer multipliers

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps that have an integer multiplier at each cycle.

Twisted linearized Reed-Solomon codes: A skew polynomial framework

We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction, which also provides a family of maximum sum-rank distance codes. As a byproduct, in the extremal case of the Hamming metric, we obtain a new family of additive MDS codes over quadratic fields.

Congruent numbers and lower bounds on class numbers of real quadratic fields

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fieldsQ(t4−n2)\mathbb {Q}(\sqrt {t^4-n^2})asttvaries over positive integers for a congruent numbernn. Furthermore, we provide lower bounds on class numbers ofRichaud-Degert typereal quadratic fields of the formQ(n2k4−1)\mathbb {Q}(\sqrt {n^2k^4-1})for positive integerskkand congruent numbersnnwhose elliptic curves have algebraic rank greater than22.