Number Of Fields Research Articles

On Σ-presentations of the additive group of the real numbers

We prove the existence of 2ω pairwise non-Σ-embeddable into each other (and henceforth non-Σ-isomorphic) Σ-presentations of the additive group of the real numbers in the hereditarily finite superstructure over the ordered field of the real numbers.

GENERALISED QUADRATIC FORMS OVER TOTALLY REAL NUMBER FIELDS

Abstract We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.

Discriminant and integral basis of number fields defined by exponential Taylor polynomials

Abstract Let $K_n=\mathbb{Q}(\alpha_n)$ be a family of algebraic number fields where $\alpha_n\in \mathbb{C}$ is a root of the nth exponential Taylor polynomial $\frac{x^n}{n!}+ \frac{x^{n-1}}{(n-1)!}+ \cdots +\frac{x^2}{2!}+\frac{x}{1!}+1$ , $n\in \mathbb{N}$ . In this paper, we give a formula for the exact power of any prime p dividing the discriminant of Kn in terms of the p-adic expansion of n. An explicit p-integral basis of Kn is also given for each prime p. These p-integral bases quickly lead to the construction of an integral basis of Kn.

On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations

AbstractIn this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily $$K_\infty $$ K ∞ -finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a “smooth version” of the space of automorphic forms, whose internal structure was the topic of Franke’s famous paper (Ann Sci de l’ENS 2:181–279, 1998). We prove that the important decomposition along the parabolic support, and the even finer—and structurally more important—decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765–790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.

Zeros of Rankin–Selberg L-functions in families

Let $\mathfrak {F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm {GL}_n$ with unitary central character over a number field $F$ . We prove the first unconditional zero density estimate for the set $\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg $L$ -functions, where $\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at $s=\frac {1}{2}$ for almost all $L(s,\pi \times \pi ')\in \mathcal {S}$ ; (ii) a strong on-average form of effective multiplicity one for almost all $\pi \in \mathfrak {F}_n$ ; and (iii) a positive level of distribution for $L(s,\pi \times \widetilde {\pi })$ , in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$ .

Slope Boundedness and Equidistribution Theorem

Abstract In this article, we prove the boundedness of minimal slopes of adelic line bundles over function fields of characteristic $0$. This can be applied to prove the equidistribution of generic and small points with respect to a big and semipositive adelic line bundle. Our methods can be applied to the finite places of number fields as well. We also show the continuity of $\chi $-volumes over function fields.

A note on quadratic forms

AbstractFor a field extension we consider maps that are quadratic over but whose polarisation is only bilinear over . Our main result is that all such are automatically quadratic forms over in the usual sense if and only if is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.

The Weyl law for congruence subgroups and arbitrary K∞-types

Let G be a reductive algebraic group over Q and Γ⊂G(Q) an arithmetic subgroup. Let K∞⊂G(R) be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of G(R) in L2(Γ﹨G(R)) of a fixed K∞-type σ. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type σ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about L-functions occurring in the constant terms of Eisenstein series. If G satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

Non-abelian p -adic Rankin-Selberg L -functions and non-vanishing of central L -values

abstract: We construct $p$-adic $L$-functions for torsion classes for $\GL(n+1)\times\GL(n)$ and along the way prove new congruences between special values of Rankin-Selberg $L$-functions for $\GL(n+1)\times\GL(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\GL(n\!+\!1)\linebreak\times\GL(n)$. As an application, we establish generic non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

Abstract Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m<n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.

General fractal dimensions of graphs of products and sums of continuous functions and their decompositions

This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type ∑ih−1(sg(ri)) for some prescribed functions h, g and for all positive real s. Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field R. Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

Torsion phenomena for zero-cycles on a product of curves over a number field

For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product X=C1×⋯×Cd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=C_1\ imes \\cdots \ imes C_d$$\\end{document} of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X. For a product X=C1×C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=C_1\ imes C_2$$\\end{document} of two curves 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} with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map J1(Q)⊗J2(Q)→εCH0(C1×C2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_1(\\mathbb {Q})\\otimes J_2(\\mathbb {Q})\\xrightarrow {\\varepsilon }{{\\,\ extrm{CH}\\,}}_0(C_1\ imes C_2)$$\\end{document} is finite, where Ji\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_i$$\\end{document} is the Jacobian variety of Ci\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_i$$\\end{document}. Our constructions include many new examples of non-isogenous pairs of elliptic curves E1,E2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_1, E_2$$\\end{document} with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products X=C1×⋯×Cd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=C_1\ imes \\cdots \ imes C_d$$\\end{document} for which the analogous map ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} has finite image.

Explicit bounds for the solutions of superelliptic equations over number fields

Abstract Let f be a polynomial with coefficients in the ring O S {O_{S}} of S-integers of a number field K, b a non-zero S-integer, and m an integer ≥ 2 {\geq 2} . We consider the following equation ( ⋆ ) {(\star)} : f ⁢ ( x ) = b ⁢ y m {f(x)=by^{m}} in x , y ∈ O S {x,y\in O_{S}} . Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of K , S , f , m {K,S,f,m} and the S-norm of b for the heights of the solutions x of equation ( ⋆ ) {(\star)} . Further, we give an explicit bound C in terms of K , S , f {K,S,f} and the S-norm of b such that if m > C {m>C} equation ( ⋆ ) {(\star)} has only solutions with y = 0 {y=0} or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of Bérczes, Evertse, and Győry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the S-norm of b instead of its height.

Entropy and energy transfer analysis of a Maxwell thin‐film fluid over an inclined surface with viscous dissipation effect

AbstractNon‐Newtonian fluid plays a vital role in the field of manufacturing and engineering sector, because of its immense heat transfer rate. The two‐dimensional incompressible unsteady Maxwell nanofluid thin‐film flow with MHD and viscous dissipation over an inclined surface is investigated. One of the mechanisms in the second law of thermodynamics, that is irreversibility which also analyzed. The Maxwell nanofluid thin film energy and motion equations in the additional information have been turned into a coupled differential system of the third order. The transformed ODE equations are further evaluated by using the Homotopy Analysis Method (HAM). The variations of entropy rate, velocity, Bejan number and temperature field with the various emerging parameters are analyzed. This could potentially lead to the development of solar energy systems that are both affordable and highly efficient, so enabling the more efficient use of renewable energy sources and reducing our dependence on fossil fuels.

On arithmetic quotients of the group SL2 over a quaternion division k-algebra

Abstract Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces X G / Γ {X_{G}/\Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k-group G = SL M 2 ⁢ ( D ) {G=\mathrm{SL}_{M_{2}(D)}} , attached to a quaternion division k-algebra D. The group G is k-simple, of k-rank one, and non-split over k. Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ {Y_{\Gamma}\subset X_{G}/\Gamma} such that its closure Y ¯ Γ {\overline{Y}_{\Gamma}} is a compact manifold with boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} , and the inclusion Y ¯ Γ → X G / Γ {\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma} is a homotopy equivalence. The connected components Y [ P ] {Y^{[P]}} of the boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k-subgroups of G. We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k-algebra D is totally definite, that is, D ramifies at all archimedean places of k, we prove that the basis of this bundle is homeomorphic to the torus T s - 1 {T^{s-1}} of dimension s - 1 {s-1} , has the compact fibre T 4 ⁢ s {T^{4s}} , and its structure group is SL 4 ⁢ s ⁢ ( ℤ ) {\mathrm{SL}_{4s}(\mathbb{Z})} . We determine the cohomology of Y [ P ] {Y^{[P]}} . Secondly, if the quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ {v\in V_{k,\infty}} at which D v {D_{v}} splits over ℝ {\mathbb{R}} , that is, D v ≅ M 2 ⁢ ( ℝ ) {D_{v}\cong M_{2}(\mathbb{R})} , the fibre is homeomorphic to T 4 ⁢ s {T^{4s}} , but the base space of the bundle is more complicated.

Density questions in rings of the form 𝒪K[γ] ∩ K

We fix a number field [Formula: see text] and study statistical properties of the ring [Formula: see text] as [Formula: see text] varies over algebraic numbers of a fixed degree [Formula: see text]. Given [Formula: see text], we explicitly compute the density of [Formula: see text] for which [Formula: see text] and show that this does not depend on the number field [Formula: see text]. In particular, we show that the density of [Formula: see text] for which [Formula: see text] is [Formula: see text]. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define [Formula: see text] to be a certain finite subset of [Formula: see text] and show that [Formula: see text] determines the ring [Formula: see text]. We show that if [Formula: see text] satisfy [Formula: see text], then the events [Formula: see text] and [Formula: see text] are independent. As [Formula: see text], we study the asymptotics of the density of [Formula: see text] for which [Formula: see text].

Reasoning Skills in Mathematics Teaching: A Meta-Synthesis on Studies Conducted in Turkey

This research aims to analyze the results of studies conducted in Turkey on reasoning skills in mathematics teaching and to reveal what kind of trend there is in this field. Within the scope of this study, databases were searched with the keywords "reasoning"(muhakeme, akıl yürütme) and "reasoning skill" (Muhakeme becerisi, akıl yürütme becerisi), and the results were examined in accordance with the inclusion criteria regarding mathematics teaching. One hundred sixty-three studies were included. Each of the studies included in the meta-synthesis study was analyzed descriptively according to type, year, method, sample type and size, data collection tools, statistical analysis, learning field, keywords, reasoning type and purpose. In addition, the studies’ results were content analyzed and tabulated by coding the differences and similarities between them with a holistic approach. It has been determined that studies on the learning fields in the mathematics curriculum are mostly carried out in the field of learning numbers and operations in the secondary school mathematics curriculum. When evaluated in terms of reasoning types, almost half of the studies were conducted on mathematical reasoning. It has been observed that after mathematical reasoning, the most focus is on proportional reasoning. When the aims of the studies included in the research were examined, it was determined that the most focused ones were "examining the factors affecting reasoning skills", "measurement of reasoning skills" and "the effect of teaching practices on reasoning skills". In the studies examined, it was seen that there were 33 teaching practices whose effects on reasoning skills were examined.

Generalized Pohst inequality and small regulators

Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over a hypercube. In this paper, we provide new and effective upper bounds for the case of fields with one complex embedding and degree between five and nine: this is done by adapting the strategy we have adopted to study the totally real case, but for this new setting several new computational issues had to be overcome. As a consequence, we detect the four number fields of signature ( r 1 , r 2 ) = ( 6 , 1 ) (r_1,r_2)=(6,1) with smallest regulator; we also expand current lists of number fields with small regulator in signatures ( 3 , 1 ) (3,1) , ( 4 , 1 ) (4,1) and ( 5 , 1 ) (5,1) .

Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2×2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results.

Hilbert’s tenth problem in anticyclotomic towers of number fields

Let K K be an imaginary quadratic field and p p be an odd prime which splits in K K . Let E 1 E_1 and E 2 E_2 be elliptic curves over K K such that the Gal ⁡ ( K ¯ / K ) \operatorname {Gal}(\bar {K}/K) -modules E 1 [ p ] E_1[p] and E 2 [ p ] E_2[p] are isomorphic. We show that under certain explicit additional conditions on E 1 E_1 and E 2 E_2 , the anticyclotomic Z p \mathbb {Z}_p -extension K anti K_{\operatorname {anti}} of K K is integrally diophantine over K K . When such conditions are satisfied, we deduce new cases of Hilbert’s tenth problem. In greater detail, the conditions imply that Hilbert’s tenth problem is unsolvable for all number fields that are contained in K anti K_{\operatorname {anti}} . We illustrate our results by constructing an explicit example for p = 3 p=3 and K = Q ( − 5 ) K=\mathbb {Q}(\sqrt {-5}) .