Algebraic Number Theory Research Articles

Arithmetic branching law and generic 𝐿-packets

Let G G be a classical group defined over a local field F F of characteristic zero. For any irreducible admissible representation π \pi of G ( F ) G(F) , which is of Casselman-Wallach type if F F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F F . In particular, if π \pi has a generic local L L -parameter, we introduce the spectral first occurrence index f s ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi ) and the arithmetic first occurrence index f a ( π ) {\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) of π \pi and prove in this paper that f s ( π ) = f a ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi )={\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) . Based on the theory of consecutive descents of enhanced L L -parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L L -packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality.

Computing superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group

In algebraic geometry or number theory, enumerating or finding superspecial curves in positive characteristic p is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group. By executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for all primes p with 19≤p<500, and in finding a single one for all primes p with 19≤p<7000.

A Study on the Classification of All Simple Commutative Near-Rings upto Isomorphism

This research paper aims to classify all simple commutative Near-Rings up to isomorphism, which are rings with no zero divisors and no ideals containing a nonzero element that is not prime ideal. The authors use various techniques from algebraic number theory and representation theory of semi simple algebras to construct representations of these rings over the integers or rational numbers. They also provide examples showing how these classes can be distinguished by their algebraic properties, such as whether they are PID (prime idempotent domain), zero-divisorless, or have ideals containing elements with prime index. The study contributes new results and insights into the classification of near-rings, which has implications for other areas of abstract algebra and number theory.

Monogenity and Power Integral Bases: Recent Developments

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled “Monogenity and Power Integral Bases”. We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given.

Monogenity of iterates of irreducible binomials

One of the oldest problems of algebraic number theory is to find a method to determine if the ring of integers of a number field K is Z [ θ ] for some θ ∈ K ; a field for which the answer to this question is affirmative is referred to as a monogenic field. Suppose f ( x ) = x m − a ∈ Z [ x ] is a monic irreducible polynomial and α n ∈ C is a root of f n ( x ) , the n-fold composition of f. In this article, we prove a necessary and sufficient condition for K n = Q ( α n ) to be monogenic for every n ∈ N and m ≥ 2.

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.

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.

Determining the primes of bad reduction of cm curves of genus 3

Abstract In this paper, we introduce a new problem called the Isogenous Embedding Problem (IEP) and relate the existence of solutions to this problem to the primes of bad reduction of complex multiplication curves of genus 3. More precisely, the absence of solutions to IEP implies potentially good reduction. We propose an algorithm for computing the solutions to the IEP and run the algorithm through different families of curves. Using this algorithm, we were able to prove the reduction type of some particular curves at certain primes that were open cases in R. Lercier, Q. Liu, E. Lorenzo García and C. Ritzenthaler, Reduction type of smooth plane quartics, Algebra Number Theory 15 no. 6 2021, 1429–1468.

ENERGY BOUNDS FOR MODULAR ROOTS AND THEIR APPLICATIONS

Abstract We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

The development of P-adic numbers theoretically and their use in number theory

Stemming from the need to formalize the divisibility of rational numbers, p-adics provide a unique mathematical perspective rooted in prime number theory. This article provides a comprehensive discussion of p-adic numbers, including their research background, definition, properties, theory, and extensions. This study first elucidates the historical background and significance of p-adic numbers, emphasizing their key role in number theory and its applications. At the heart of this research is a deep dive into the precise definition of p-radical numbers, revealing their unique, often counterintuitive, distance measure. We look at their fundamental characteristics, demonstrate their illogical characteristics, and discuss how they affect transcendental number theory, Diophantine equations, and algebraic number theory. Furthermore, this article explores the theoretical foundations of p-radical numbers and their extensions, emphasizing their integral role in advanced mathematical structures such as p-radical analysis and p-radical geometry. By covering these aspects, this study aims to highlight the lasting impact of p-adic on modern mathematics, reshape our understanding of divisibility, and advance mathematical inquiry into new and uncharted territory.

Algebraic, Analytic, and Computational Number Theory and Its Applications

Analytic number theory is a branch of number theory which inherits methods from mathematical analysis in order to solve difficult problems about the integers [...]

The interconnection between local and global m-th powers and Grunwald-Wang theorem

The relation between local and global solution of an equation can be discussed with the method of class field theory and algebraic number theory. In this piece of writing, the author will introduce the behavior of both local and global m-th power in some specific number field. Of course, the result in this paper can be extended into the function field, but it will not be involved in this paper. This paper will prove that if k(_(2^t ))/k is cyclic then the Local m-th powers everywhere is equivalent to the global m-th power. In the Non-cyclic case this decomposition becomes P(m,S)=k^mk^m. This paper will also prove some useful propositions in topological group theory, which will be used in the proof of Grunwald-Wang theorem. Grunwald-Wang theorem states that we can find a cyclic extension with given local behaviors. To describe the extension, this paper combines character theory with a topological group, one can depict the cyclic extension. This theorem can be used in the further exploration of central simple algebra.

The special value 𝑢=1 of Artin-Ihara 𝐿-functions

We study the special value u = 1 u=1 of Artin-Ihara L L -functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.

On Algebraic Number Theory Domains Dedekind

This article is based on a series talks I gave on the topic mentioned in the title in a Counsellor Seminar at proms’. The exposition till Section is based on and most of the remainder is based on and some notes on Algebraic Number Theory available.

ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS

AbstractLet $K={\mathbb {Q}}(\theta )$ be an algebraic number field with $\theta $ satisfying a monic irreducible polynomial $f(x)$ of degree n over ${\mathbb {Q}}.$ The polynomial $f(x)$ is said to be monogenic if $\{1,\theta ,\ldots ,\theta ^{n-1}\}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc.100 (2019), 239–244] conjectured that if $n\ge 3$ , $1\le m\le n-1$ , $\gcd (n,mB)=1$ and A is a prime number, then the polynomial $x^n+A (Bx+1)^m\in {\mathbb {Z}}[x]$ is monogenic if and only if $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ is square-free. We prove that this conjecture is true.

ℤ-gradings on the Grassmann algebra over infinite fields: Graded identities and central polynomials

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of E equipped with its [Formula: see text] and 3-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.

Extended higher Herglotz functions I. Functional equations

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F(x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F(x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study Fk,N(x), an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by Fk,N(x). Radchenko and Zagier gave a beautiful relation between the integral ∫01log⁡(1+tx)1+tdt and F(x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between Fk,N(x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of Fk,N(x) and some generalized Lambert series are also obtained along with other supplementary results.

A Brief Introduction to Splitting Of Primes over Number Fields

The study of Dedekind Zeta Functions over a number field extension uses different aspects of both Algebraic and Analytic Number Theory. In this paper, we shall learn about the structure and different analytic aspects of such functions, namely the domain of its convregence and analyticity at different points of ℂ when the function is defined over any finite field extension K over ℚ . Moreover, given any two Number Fields L and K over ℚ with L being Normal over K, our intention is to classify and study the primes in K which split completely in L. Also, we shall explore some special cases related to this result

A functor for constructing R$R$‐matrices in the category O$\mathcal {O}$ of Borel quantum loop algebras

AbstractWe tackle the problem of constructing ‐matrices for the category associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra . For this, we define an invertible exact functor from the category linked to to the one linked to . This functor is compatible with tensor products, preserves irreducibility, and interchanges the subcategories and of Hernandez and Leclerc (Algebra Number Theory 10 (2016) 2015–2052). We construct ‐matrices for by applying on the braidings already found for by Hernandez (Represent. Theory 26 (2022) 179–210). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring as well as a functorial interpretation of a remarkable ring isomorphism of Hernandez–Leclerc.

History and development of algebraic number theory

We briefly describe the origin and development of classical Algebraic Number Theory. We state some latest results regarding discriminants and integral bases of pure number fields as well as sextic number fields defined by irreducible trinomials with integer coefficients. A few open problems regarding class number of algebraic number fields are also mentioned.