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Axioms
Special Issues
Elliptic Curves, Modular Forms, L-Functions and Applications
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Elliptic Curves, Modular Forms, L-Functions and Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 27 June 2025 | Viewed by 826

Special Issue Editors

Dr. Dongxi Ye

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Guest Editor
School of Mathematics, Sun Yat-sen University, Zhuhai, China
Interests: number theory; modular forms
Dr. Fabio Caldarola

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Guest Editor
Department of Mathematics and Computer Science, Cubo 31/A, Università della Calabria, 87036 Rende, Italy
Interests: number theory; iwasawa theory; combinatorics; fibonacci numbers; mumber sequences; graph theory; unimaginable numbers; combinatorics on words; fractal geometry; polytopes; elliptic curves; cryptography; applied mathematics; cellular automata; mathematical models; chaos theory; nonlinear dynamics; shallow water
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue of Axioms focuses on the intricate and profound connections between elliptic curves, modular forms, their associated L-functions, and related topics. These areas represent the fundamental pillars of modern number theory, with far-reaching implications in pure and applied mathematics, such as combinatorics, cryptography, coding theory, mathematical physics, etc.

The scope of this Special Issue encompasses theoretical advancements, computational breakthroughs, and the interdisciplinary applications of these topics. Areas of interest include, but are not limit to, the arithmetic of elliptic curves, modular forms and their generalizations, analytic and algebraic properties of L-functions, and applications in number theory and combinatorics.

The purpose of this Special Issue is to provide a platform for the dissemination of cutting-edge research that deepens our understanding of elliptic curves, modular forms, and L-functions while fostering dialogue among researchers across different mathematical disciplines. By bringing together a diverse collection of original research papers and surveys, this Special Issue aims to achieve the following:

  1. Highlight recent developments in the theory of elliptic curves, including, but not limit to, the Birch and Swinnerton-Dyer conjecture, the rank of elliptic curves, and their rational points.
  2. Explore the role of modular forms and their generalizations in arithmetic geometry, number theory, and combinatorics.
  3. Examine the analytic and arithmetic properties of L-functions, particularly their connections to special values, algebraic cycles, etc.
  4. Showcase innovative computational approaches and algorithms that enhance the understanding and application of these mathematical structures.
  5. Encourage interdisciplinary research by highlighting the applications of elliptic curves, modular forms, and L-functions in cryptography, coding theory, and other areas.

Dr. Dongxi Ye
Prof. Dr. Fabio Caldarola
Guest Editors

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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • combinatorics
  • elliptic curves
  • L-functions
  • modular forms
  • number theory

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (2 papers)

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Research

11 pages, 278 KiB  
Article
On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields
by Abdulaziz Deajim
Axioms 2025, 14(5), 335; https://doi.org/10.3390/axioms14050335 - 27 Apr 2025
Viewed by 84
Abstract
Consider the modular group PSL(2,Z)=⟨x,y|x2=y3=1⟩ generated by the transformations x:z↦−1/z and [...] Read more.
Consider the modular group PSL(2,Z)=⟨x,y|x2=y3=1⟩ generated by the transformations x:z↦−1/z and y:z↦(z−1)/z. Let H be the proper subgroup ⟨y,v|y3=v3=1⟩ of PSL(2,Z), where v=xyx. For a positive square-free integer n, this article studies the action of H on the subset {a+−nc|a,b=a2+nc,c∈Z,c≠0} of the imaginary quadratic number field Q(−n) where, in particular, the accurate estimate of the number of orbits arising from this action is given, correcting the estimate given in some of the relevant literature. Full article
(This article belongs to the Special Issue Elliptic Curves, Modular Forms, L-Functions and Applications)
12 pages, 261 KiB  
Article
Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis
by Emilio R. Negrín, Jeetendrasingh Maan and Benito J. González
Axioms 2025, 14(2), 129; https://doi.org/10.3390/axioms14020129 - 10 Feb 2025
Cited by 1 | Viewed by 494
Abstract
This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we [...] Read more.
This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms. Full article
(This article belongs to the Special Issue Elliptic Curves, Modular Forms, L-Functions and Applications)
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