Talteori

Den græske matematiker Euklids lærebog ELEMENTER er, hvis man ser bort fra Bibelen, den mest trykte lærebog gennem tiderne. ELEMENTER er skrevet omkring 300 f. Kristus, og er en samling af i alt 13 bøger, der præsenterer den samlede viden om matematik og geometri i tiden.ELEMENTER er et helt uomgængeligt værk for den matematik- og filosofiinteresserede.Bøgerne 1-6, der behandler plangeometrien, er udgivet i foreløbig to oplag, og nu udgives bøgerne 7-9, der omhandler naturlige tal, som bind 2 i det samlede værk.

Bind 3 i den nye oversættelse af den græske matematiker Euklids lærebog ELEMENTER omhandler kommensurabilitet. ELEMENTER er skrevet omkring 300 f. Kristus og er en samling af i alt 13 bøger, der præsenterer den samlede viden om matematik og geometri i tiden.Engang omkring år 500 f.Kr. opdagede nogle pythagoræiske matematikere et matematisk problem, som de meget tøvende gik i gang med at finde løsninger på: Nogle tal og linjestykker kan måles med den samme målestok, andre kan ikke.  Et kendt eksempel er et kvadrat på 2x2 fod (eller meter), hvis diagonal målenheden ikke ’går op’ i.Linjer er med andre ord kommensurable eller inkommensurable med hinanden. Men hvordan hænger det sammen?I de foregående ni bøger i Elementerne har Euklid fremstillet geometrien i sin klassiske form på en ny og overskuelig måde, der gjorde alle tidligere geometrier forældede, men i Bog X tager han fat på læsningerne i en helt ny og uudforsket gren af matematikken, læren om kommensurabilitet, som havde fået et afgørende spring fremad med matematikerne Theodoros og Theaitetos i begyndelsen af 300-tallet f.Kr., men endnu ikke var blevet lærebogsstof. Bog X, som i omfang udgør næsten en fjerdedel af hele Euklids Elementer, bærer som frontforskning præg af at skulle fremstille en ny og uprøvet videnskabsgren i et forsøg på at bringe orden og system i den nye erkendelse.Euklids ELEMENTER er et hovedværk ikke alene i matematikken, men i hele den vestlige civilisation og er et helt uomgængeligt værk for den matematik- og filosofiinteresserede.Bogen henvender sig til en bred gruppe læsere: professionelle matematikere fx i gymnasiet og på universitetet, hvor der undervises i matematikkens historie og videnskabsteori, filosofi- og antikstuderende sammesteds og endelig den ret store gruppe mennesker, der interesserer sig for antik filosofi og matematik.Euklids ELEMENTER bog X er indledt, oversat og forsynet med bemærkninger af Claus Glunk, Hanne Eggert Strand, Chr. Marinus Taisbak og Chr. Gorm Tortzen.

This book serves as an illuminating introduction to the intricacies of the geometry of numbers. It commences by exploring basic concepts of convex sets and lattices in Euclidean space and goes on to delve into Minkowski¿s fundamental theorem for convex bodies and its applications. It discusses critical determinants and successive minima before explaining the core results of packings and coverings. The text goes on to delve into the significance of renowned conjectures such as Minkowski¿s conjecture regarding the product of linear forms, Watson¿s conjecture, and the conjecture of Bambah, Dumir, and Hans-Gill concerning non-homogeneous minima of indefinite quadratic forms. Dedicated to Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India, this comprehensive book addresses both homogeneous and non-homogeneous problems, while sprinkling in historical insights and highlighting unresolved questions in the field. It is ideally suited for beginnersembarking on self-study as well as for use as a text for a one- or two-semester introductory course.

This book describes a novel approach to the study of Siegel modular forms of degree two with paramodular level. It introduces the family of stable Klingen congruence subgroups of GSp(4) and uses this family to obtain new relations between the Hecke eigenvalues and Fourier coefficients of paramodular newforms, revealing a fundamental dichotomy for paramodular representations. Among other important results, it includes a complete description of the vectors fixed by these congruence subgroups in all irreducible representations of GSp(4) over a nonarchimedean local field.Siegel paramodular forms have connections with the theory of automorphic representations and the Langlands program, Galois representations, the arithmetic of abelian surfaces, and algorithmic number theory. Providing a useful standard source on the subject, the book will be of interest to graduate students and researchers working in the above fields.

This is a collection of lecture notes from the minicourses in the December 2018 Langlands Workshop: Endoscopy and Beyond. The volume combines seven introductory chapters on trace formulas, local Arthur packets, and beyond endoscopy. It aims to introduce the endoscopy classification via a basic example of the trace formula for SL(2), explore the more refined questions on the structure of Arthur packets, and look beyond endoscopy following the suggestions of Langlands, Braverman-Kazhdan, Ngo, and Altü. The book is a helpful reference for undergraduates, postgraduates, and researchers.

Dieses Buch versucht, die schrittweise Entwicklung der wichtigsten Forschungsinstitute zur Zahlentheorie in Südindien, Punjab, Mumbai, Bengalen und Bihar zu beschreiben, einschließlich der Gründung des Tata Institute of Fundamental Research (TIFR) in Mumbai, einem bahnbrechenden Ereignis in der Geschichte der Zahlentheorie-Forschung in Indien. Die Forschung zur Zahlentheorie in Indien begann in der modernen Zeit mit dem Auftreten des ikonischen Genies Srinivasa Ramanujan, das Mathematiker auf der ganzen Welt inspirierte. Das Buch diskutiert die nationale und internationale Wirkung der Forschung indischer Zahlentheoretiker und enthält eine sorgfältig zusammengestellte, umfassende Bibliographie bedeutender indischer Zahlentheoretiker des 20. Jahrhunderts. Es ist wichtig für die historische Dokumentation und eine wertvolle Ressource für Forscher auf diesem Gebiet. Das Buch diskutiert auch kurz die Bedeutung der Zahlentheorie in der modernen Mathematik, einschließlich Anwendungen der Ergebnisse indigener Zahlentheoretiker in praktischen Bereichen. Da das Buch aus der Perspektive der Wissenschaftsgeschichte geschrieben ist, wurden technische Fachbegriffe und mathematische Ausdrücke so weit wie möglich vermieden.Die Übersetzung wurde mit Hilfe von künstlicher Intelligenz durchgeführt. Eine anschließende menschliche Überarbeitung erfolgte vor allem in Bezug auf den Inhalt.

This book provides a foundation for arithmetic topology, a new branch of mathematics that investigates the analogies between the topology of knots, 3-manifolds, and the arithmetic of number fields. Arithmetic topology is now becoming a powerful guiding principle and driving force to obtain parallel results and new insights between 3-dimensional geometry and number theory.After an informative introduction to Gauss' work, in which arithmetic topology originated, the text reviews a background from both topology and number theory. The analogy between knots in 3-manifolds and primes in number rings, the founding principle of the subject, is based on the étale topological interpretation of primes and number rings. On the basis of this principle, the text explores systematically intimate analogies and parallel results of various concepts and theories between 3-dimensional topology and number theory. The presentation of these analogies begins at an elementary level, gradually building to advanced theories in later chapters. Many results presented here are new and original.References are clearly provided if necessary, and many examples and illustrations are included. Some useful problems are also given for future research. All these components make the book useful for graduate students and researchers in number theory, low dimensional topology, and geometry.This second edition is a corrected and enlarged version of the original one. Misprints and mistakes in the first edition are corrected, references are updated, and some expositions are improved. Because of the remarkable developments in arithmetic topology after the publication of the first edition, the present edition includes two new chapters. One is concerned with idelic class field theory for 3-manifolds and number fields. The other deals with topological and arithmetic Dijkgraaf¿Witten theory, which supports a new bridge between arithmetic topology and mathematical physics.

This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the ¿abelian¿ Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the ¿non-abelian¿ modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included.These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms.Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.