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Mathematics
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New Developments in Number Theory
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New Developments in Number Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (10 February 2022) | Viewed by 8601

Special Issue Editor

Prof. Dr. Luis Dieulefait

E-Mail Website
Guest Editor
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain
Interests: number theory; arithmetic geometry

Special Issue Information

Dear Colleagues,

In this Special Issue, we would like to include recent developments in several branches of number theory, including arithmetic geometry, the theory of modular and automorphic forms and the Langlands program, analytic number theory, algebraic number theory, Galois theory, Arakelov geometry, Diophantine equations and applications to cryptography. This century has already seen several remarkable results in each and every one of these branches, and current research will certainly lead to new exciting developments, some of which we would like to include in this Special Issue. We welcome research papers on both theoretical and computational aspects of each of the subjects described.

Prof. Dr. Luis Dieulefait
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Arithmetic geometry
  • Elliptic curves
  • Modular forms
  • Automorphic forms
  • Galois theory
  • Field theory
  • Algebraic number theory
  • Analytic number theory
  • Sieve theory
  • Langlands program
  • Inverse Galois problem
  • Finite fields
  • Arakelov geometry
  • Additive number theory
  • Cryptography
  • Diophantine equations

Published Papers (5 papers)

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Research

10 pages, 299 KiB  
Article
Seven Small Simple Groups Not Previously Known to Be Galois Over Q
by Luis Dieulefait, Enric Florit and Núria Vila
Mathematics 2022, 10(12), 2048; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122048 - 13 Jun 2022
Viewed by 1281
Abstract
In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of [...] Read more.
In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms. Full article
(This article belongs to the Special Issue New Developments in Number Theory)
12 pages, 256 KiB  
Article
Square Integer Matrix with a Single Non-Integer Entry in Its Inverse
by Arif Mandangan, Hailiza Kamarulhaili and Muhammad Asyraf Asbullah
Mathematics 2021, 9(18), 2226; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182226 - 10 Sep 2021
Viewed by 1261
Abstract
Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, [...] Read more.
Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix. Full article
(This article belongs to the Special Issue New Developments in Number Theory)
13 pages, 284 KiB  
Article
On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins
by Xue Han, Xiaofei Yan and Deyu Zhang
Mathematics 2021, 9(11), 1254; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111254 - 30 May 2021
Cited by 4 | Viewed by 1664
Abstract
Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier [...] Read more.
Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group. Full article
(This article belongs to the Special Issue New Developments in Number Theory)
6 pages, 253 KiB  
Article
Schanuel’s Conjecture and the Transcendence of Power Towers
by Eva Trojovská and Pavel Trojovský
Mathematics 2021, 9(7), 717; https://0-doi-org.brum.beds.ac.uk/10.3390/math9070717 - 26 Mar 2021
Viewed by 1862
Abstract
We give three consequences of Schanuel’s Conjecture. The first is that P(e)Q(e) and P(π)Q(π) are transcendental, for any non-constant polynomials [...] Read more.
We give three consequences of Schanuel’s Conjecture. The first is that P(e)Q(e) and P(π)Q(π) are transcendental, for any non-constant polynomials P(x),Q(x)∈Q¯[x]. The second is that π≠αβ, for any algebraic numbers α and β. The third is the case of the Gelfond’s conjecture (about the transcendence of a finite algebraic power tower) in which all elements are equal. Full article
(This article belongs to the Special Issue New Developments in Number Theory)
26 pages, 387 KiB  
Article
Extremal p-Adic L-Functions
by Santiago Molina
Mathematics 2021, 9(3), 234; https://0-doi-org.brum.beds.ac.uk/10.3390/math9030234 - 25 Jan 2021
Viewed by 1221
Abstract
In this note, we propose a new construction of cyclotomic p-adic L-functions that are attached to classical modular cuspidal eigenforms. This allows for us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic [...] Read more.
In this note, we propose a new construction of cyclotomic p-adic L-functions that are attached to classical modular cuspidal eigenforms. This allows for us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic forms on arbitrary groups. In the classical setting of GL2 over Q, this allows for us to construct the p-adic L-function in the so far uncovered extremal case, which arises under the unlikely hypothesis that p-th Hecke polynomial has a double root. Although Tate’s conjecture implies that this case should never take place for GL2/Q, the obvious generalization does exist in nature for Hilbert cusp forms over totally real number fields of even degree, and this article proposes a method that should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functionsLpext(f,s), and relate Lpext(f,s) to the two-variable p-adic L-function interpolating cyclotomic p-adic L-functions along a Coleman family. Full article
(This article belongs to the Special Issue New Developments in Number Theory)
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