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2019 | Book

An Introduction to Modular Forms Read chapter Read first chapter

Notes from the International Autumn School on Computational Number Theory

Editors: Dr. Ilker Inam, Engin Büyükaşık

Publisher: Springer International Publishing

Book Series : Tutorials, Schools, and Workshops in the Mathematical Sciences

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume collects lecture notes and research articles from the International Autumn School on Computational Number Theory, which was held at the Izmir Institute of Technology from October 30th to November 3rd, 2017 in Izmir, Turkey. Written by experts in computational number theory, the chapters cover a variety of the most important aspects of the field. By including timely research and survey articles, the text also helps pave a path to future advancements. Topics include:Modular forms
L-functions
The modular symbols algorithm
Diophantine equations
Nullstellensatz
Eisenstein series
Notes from the International Autumn School on Computational Number Theory will offer graduate students an invaluable introduction to computational number theory. In addition, it provides the state-of-the-art of the field, and will thus be of interest to researchers interested in the field as well.

Table of Contents

Frontmatter

Lecture Notes

Frontmatter
An Introduction to Modular Forms
Abstract
In this course, we introduce the main notions relative to the classical theory of modular forms. A complete treatise in a similar style can be found in the author’s book joint with Strömberg (Cohen and Strömberg, Modular Forms: A Classical Approach, Graduate Studies in Math. 179, American Math. Soc. (2017) [1]).
Henri Cohen
Computational Arithmetic of Modular Forms
Abstract
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided.
Gabor Wiese
Computational Number Theory in Relation with L-Functions
Abstract
We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance, Dirichlet L-functions, involving in particular the study of inverse Mellin transforms); we also give a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions.
Henri Cohen
Exponential Diophantine Equations
Abstract
This paper is a very gentle introduction to solving exponential Diophantine equations using the technology of linear forms in logarithms of algebraic numbers.
Florian Luca

Research Contributions

Frontmatter
Nullstellensatz via Nonstandard Methods
Abstract
In this short note, we survey some degree and height bound results for arithmetic Nullstellensatz from the literature. We also introduce the notion of height functions, ultraproducts and nonstandard extensions. As our main remark, we find height bounds for polynomial rings over integral domains via nonstandard methods.
Haydar Göral
On the -Constacyclic and Cyclic Codes Over the Finite Ring
Abstract
In this paper a new finite ring is introduced along with its algebraic properties. In addition, a new Gray map is defined on the ring. The Gray images of both the cyclic and the \((1+u^{2}+u^{3})\)-constacyclic codes over the finite ring are found to be permutation equivalent to binary quasicyclic codes.
G. Gözde Güzel, Abdullah Dertli, Yasemin Çengellenmiş
On Higher Congruences Between Cusp Forms and Eisenstein Series. II.
Abstract
We study congruences between cuspidal modular forms and Eisenstein series at levels which are square-free integers and for equal even weights. This generalizes our previous results from (Naskręcki, Computations with modular forms, Contrib. Math. Comput. Sci., vol. 6, pp. 257–277. Springer, Cham (2014) [17]) for prime levels and provides further evidence for the sharp bounds obtained under restrictive ramification conditions. We prove an upper bound on the exponent in the general square-free situation and also discuss the existence of the congruences when the coefficients belong to the rational numbers and weight equals 2.
Bartosz Naskręcki
Lucas Numbers Which are Products of Two Balancing Numbers
Abstract
In this paper, we find all Lucas numbers, which are products of two balancing numbers.
Zafer Şiar
Metadata
Title
Notes from the International Autumn School on Computational Number Theory
Editors
Dr. Ilker Inam
Engin Büyükaşık
Copyright Year
2019
Electronic ISBN
978-3-030-12558-5
Print ISBN
978-3-030-12557-8
DOI
https://doi.org/10.1007/978-3-030-12558-5

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