Algebraic Number Research Articles

Co-Axial Metrics on the Sphere and Algebraic Numbers

In this paper, we consider the following curvature equation $$\Delta u+{\rm e}^u=4\pi\biggl((\theta_0-1)\delta_0+(\theta_1-1)\delta_1 +\sum_{j=1}^{n+m}\bigl(\theta_j'-1\bigr)\delta_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+\theta_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $\theta_0$, $\theta_1$, $\theta_\infty$, and $\theta_{j}'$ are positive non-integers for $1\le j\le n$, while $\theta_{j}'\in\mathbb{N}_{\geq 2}$ are integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le j\le n+m$, with angles $2\pi\theta_0$, $2\pi\theta_1$, $2\pi\theta_\infty$, and $2\pi\theta_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric ${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the projective monodromy group is contained in the unit circle. The sufficient and necessary conditions in terms of angles for the existence of such metrics were obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$. Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such that a co-axial metric exists, among other things we prove that (i) If $m=1$, i.e., there is only one integer $\theta_{n+1}'$ among $\theta_j'$, then $A$ is a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.

Research and Application of Multiwave Number Line measurement Model based on Geometric Algebraic Plane Number Equation

Research and Application of Multiwave Number Line measurement Model based on Geometric Algebraic Plane Number Equation

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

Abstract We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math.521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

A note on logarithmic equidistribution

For every algebraic number [Formula: see text] on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of [Formula: see text] at the conjugates are essentially bounded from above by [Formula: see text]. This completes a characterization on functions [Formula: see text] initiated by Autissier and Baker–Masser, who cover the cases [Formula: see text] and [Formula: see text], respectively. Using the same ideas we also prove analogues in the [Formula: see text]-adic setting.

Torsion primes for elliptic curves over degree 8 number fields

Let d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document} be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8.

Riemann zeta functions for Krull monoids

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function.

On the modulo p zeros of modular forms congruent to theta series

For a prime p larger than 7, the Eisenstein series of weight p−1 has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight k≥4 for the full modular group as the modular forms for which the first dim⁡(Mk) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

Rotation number of 2-interval piecewise affine maps

AbstractWe study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $$f_{\varvec{p}}$$ f p are parametrized by a quintuple $$\varvec{p}$$ p of real numbers satisfying inequations. Viewing $$f_{\varvec{p}}$$ f p as a circle map, we show that it has a rotation number $$\rho (f_{\varvec{p}})$$ ρ ( f p ) and we compute $$\rho (f_{\varvec{p}})$$ ρ ( f p ) as a function of $$\varvec{p}$$ p in terms of Hecke–Mahler series. As a corollary, we prove that $$\rho (f_{\varvec{p}})$$ ρ ( f p ) is a rational number when the components of $$\varvec{p}$$ p are algebraic numbers.

Partitions into powers of an algebraic number

Partitions into powers of an algebraic number

The distribution of the multiplicative index of algebraic numbers over residue classes

The distribution of the multiplicative index of algebraic numbers over residue classes

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

AbstractLet $K_n=\mathbb{Q}(\alpha_n)$ 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 solutions of some Lebesgue–Ramanujan–Nagell type equations

Denote by [Formula: see text] the class number of the imaginary quadratic field [Formula: see text] with [Formula: see text] prime. It is well known that [Formula: see text] for [Formula: see text]. Recently, all the solutions of the Diophantine equation [Formula: see text] with [Formula: see text] were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation [Formula: see text] in unknown integers [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.

Random Algebraic Numbers

Random Algebraic Numbers

Higher-degree Artin conjecture

ABSTRACT For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.

The Vladimirov operator with variable coefficients on finite adeles and the Feynman formulas for the Schrödinger equation

We construct the Hamiltonian Feynman, Lagrangian Feynman, and Feynman–Kac formulas for the solution of the Cauchy problem with the Schrödinger operator −MgDα − V, where Dα is the Vladimirov operator and Mg is the operator of multiplication by a real-valued function g defined on the d-dimensional space AKd of finite adeles over the algebraic number field K.

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.

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.

Towards a Generalized Cayley–Dickson Construction through Involutive Dimagmas

A generalized construction procedure for algebraic number systems is hereby presented. This procedure offers an efficient representation and computation method for complex numbers, quaternions, and other algebraic structures. The construction method is then illustrated across a range of examples. In particular, the novel developments reported herein provide a generalized form of the Cayley–Dickson construction through involutive dimagmas, thereby allowing for the treatment of more general spaces other than vector spaces, which underlie the associated algebra structure.

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].