DOCSLIB.ORG

Introduction to Analytic Number Theory Pdf, Epub, Ebook

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

INTRODUCTION TO ANALYTIC NUMBER THEORY PDF, EPUB, EBOOK Tom M. Apostol | 352 pages | 28 May 1998 | Springer-Verlag New York Inc. | 9780387901633 | English | New York, NY, United States Introduction to Analytic Number Theory PDF Book Please respect the publisher and the author for their creations if their books are copyrighted. Jovany Agathe rated it it was ok Feb 24, Steve rated it really liked it Aug 25, We use essential cookies to perform essential website functions, e. You can learn more about how we plus approved third parties use cookies and how to change your settings by visiting the Cookies notice. Algebraic number theory Analytic number theory Geometric number theory Computational number theory Transcendental number theory Diophantine geometry Arithmetic combinatorics Arithmetic geometry Arithmetic topology Arithmetic dynamics. The treatment is basic and understandable for those who have basic knowledge of real analysis. Welcome back. The prime number theorem can be generalised to this problem; letting. Laszlo Lovasz. Instead I'll place the following books on reserve at the library:. Table of contents 1: The Fundamental Theorem of Arithmetic. David A. Riemann's statement of the Riemann hypothesis, from his paper. If it's just the first time you approach to the subject you better look for other books as well. Serge Lang. Harry Potter. Other topics may include sieves e. Deepak rated it liked it Aug 11, Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. Want to Read saving…. Geometry: Euclid and Beyond Robin Hartshorne. In a single short paper the only one he published on the subject of number theory , he investigated the Riemann zeta function and established its importance for understanding the distribution of prime numbers. Sign up now. Accept all Manage Cookies. Nair February Gabriele Biondo rated it it was amazing Dec 29, ON OFF. If you're concerned with the background, please feel free to talk to me. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate. Numbers Natural numbers Prime numbers Rational numbers Irrational numbers Algebraic numbers Transcendental numbers p-adic numbers Arithmetic Modular arithmetic Arithmetic functions. Some examples are i Montgomery's pair correlation conjecture and the work that initiated from it, ii the new results of Goldston, Pintz and Yilidrim on small gaps between primes , and iii the Green—Tao theorem showing that arbitrarily long arithmetic progressions of primes exist. Stephen Abbott. In proving the theorem, he introduced the Dirichlet characters and L-functions. The topics chosen are carefully chosen and explicitly dealt with. Another recent development is probabilistic number theory , [11] which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has. Kristopher Tapp. Reload to refresh your session. This site does not host pdf files, does not store any files on its server, all document are the property of their respective owners. Recent Search ibc codes pdf free download mp2t pdftotiff Free pdf copy of NFPA fnaf the twisted ones book free online read edit pdf files with pdfescape an online free pdf sacred geometry deciphering the code geography of tourism free elna service manual pfmea aiag pdf wellcare catalog Number theoretic methods: future trends. Introduction to Analytic Number Theory Writer David Shaw rated it it was amazing Nov 21, Noah Drake rated it it was amazing Aug 18, Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. This remarkable result is what is now known as the prime number theorem. Close X. Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. Homework Assignments. Coronavirus delivery updates. Main article: Diophantine problem. Sorry, something went wrong. Table of contents 1: The Fundamental Theorem of Arithmetic. Tom M. The prime number theorem can be generalised to this problem; letting. In Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li x under the slightly different form of a series, which he communicated to Gauss. An important question is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. Essential We use cookies to provide our services , for example, to keep track of items stored in your shopping basket, prevent fraudulent activity, improve the security of our services, keep track of your specific preferences e. Deepak rated it liked it Aug 11, Peter J. We use cookies to improve this site Cookies are used to provide, analyse and improve our services; provide chat tools; and show you relevant content on advertising. Library 6, Trivia About Introduction to A Readers also enjoyed. Show less Show more Advertising ON OFF We use cookies to serve you certain types of ads , including ads relevant to your interests on Book Depository and to work with approved third parties in the process of delivering ad content, including ads relevant to your interests, to measure the effectiveness of their ads, and to perform services on behalf of Book Depository. Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory, [3] a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. A very good undergraduate introductory book to analytic number theory. Introduction to Analytic Number Theory Reviews Cookie Preferences We use cookies and similar tools, including those used by approved third parties collectively, "cookies" for the purposes described below. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series or a product of simpler Dirichlet series using convolution identities , examine this series as a complex function and then convert this analytic information back into information about the original function. Go to file T Go to line L Copy path. Namespaces Article Talk. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than pages. Classical theory. The prime number theorem can be generalised to this problem; letting. Rating details. Groups and Symmetry Mark A. Jun 25, Elizabeth S. This site does not host pdf files, does not store any files on its server, all document are the property of their respective owners. Open Preview See a Problem? Riemann found that the error terms in this expression, and hence the manner in which the primes are distributed, are closely related to the complex zeros of the zeta function. Some examples are i Montgomery's pair correlation conjecture and the work that initiated from it, ii the new results of Goldston, Pintz and Yilidrim on small gaps between primes , and iii the Green—Tao theorem showing that arbitrarily long arithmetic progressions of primes exist. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. I want to read this book, please! Details if other :. Much of analytic number theory was inspired by the prime number theorem. For example, we use cookies to conduct research and diagnostics to improve our content, products and services, and to measure and analyse the performance of our services. Gauss , amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number N is close to the value of the integral. Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory, [3] a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. Algebraic number theory Analytic number theory Geometric number theory Computational number theory Transcendental number theory Diophantine geometry Arithmetic combinatorics Arithmetic geometry Arithmetic topology Arithmetic dynamics. Please respect the publisher and the author for their creations if their books are copyrighted. Student Math. But Gauss never published this conjecture. Library 6, Lists with This Book. Review quote From the reviews: T. Get A Copy. Office Hours I'll be available from on Tuesday afternoons; my office is in W. Introduction to Analytic Number Theory Read Online In he published Dirichlet's theorem on arithmetic progressions , using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. Views Read Edit View history. Copyright Disclaimer: All books are the property of their respective owners. Sign up now. The fields of diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture. The general case was proved by Hilbert in , using algebraic techniques which gave no explicit bounds. For this reason, the book starts with the most elementary properties of the natural integers. Home Contact us Help Free delivery worldwide. Analytic number theorists are often interested in the error of approximations such as the prime number theorem. It's a classic, no doubt about that but, it doesn't go deep into stuff just touches then leaves the topic. Close X. Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture. Vaughan, Multiplicative number theory, I. Main article: Pafnuty Chebyshev. In particular, they proved that if. Analytics cookies We use analytics cookies to understand how you use our websites so we can make them better, e. Robin Hartshorne. Complex Analysis Joseph Bak. You can always update your selection by clicking Cookie Preferences at the bottom of the page.
Recommended publications
  • HKUST Newsletter-Genesis

    HKUST Newsletter-Genesis

    World’s first smart coating - NEWSLETTER GENESIS to fight infectious diseases Issue 8 2O1O 全球首創智能塗層 有效控制傳染病 Aeronautics guru lands on HKUST – New Provost Prof Wei Shyy 航天專家科大著陸－ 新首席副校長史維教授 Welcome 3-3-4 迎接三三四 目錄 C ontents President’s Message 校長的話 Physics professor awarded Croucher Fellowship 22 物理學系教授榮膺裘槎基金會優秀科研者 President’s Message 2 校長的話 Prof Wang Wenxiong of Biology awarded 23 First Class Prize by China’s Ministry of Education 生物系王文雄教授獲國家教育部優秀成果獎 Eureka! - Our Search & Research 一等獎 雄雞鳴 天下白―科研成果 香港―我們的家 HKUST develops world’s first smart anti-microbial Local 4 coating to control infectious diseases 科大研發全球首創智能殺菌塗層 有效控制傳染病 We are ready for 3-3-4 24 科大為迎接「三三四」作好準備 An out of this world solution to the 8 energy crisis: The moon holds the answer Legendary gymnast Dr Li Ning 月球的清潔能源可望解決地球能源危機 27 shares his dream at HKUST 與李寧博士對談：成功源自一個夢想 HKUST achieves breakthrough in wireless 10 technology to facilitate network traffic Chow Tai Fook Cheng Yu Tung Fund 科大無線通訊突破 有效管理網絡交通 28 donates $90 million to HKUST 科大喜獲周大福鄭裕彤基金捐贈九千萬元 HKUST develops hair-based drug testing 12 科大頭髮驗毒提供快而準的測試 CN Innovations Ltd donates $5 million 30 to HKUST to support applied research 中南創發有限公司捐贈五百萬元 The answer is in the genes: HKUST joins high- 助科大發展應用研究 14 powered global team to decode cancer genome 科大參與國際聯盟破解癌症基因 Honorary Fellowship Presentation Ceremony - 31 trio honored for dual achievement: doing good Raising the Bar 又上一層樓 while doing well 科大頒授榮譽院士 表揚社會貢獻 Celebrating our students’ outstanding 16 achievements 學生獲著名大學研究院錄取 師生同慶 National 祖國―我們的根 HKUST puts on great performance
  • Advances in Pure Mathematics Special Issue on Number Theory

    Advances in Pure Mathematics Special Issue on Number Theory

    Advances in Pure Mathematics Scientific Research Open Access ISSN Online: 2160-0384 Special Issue on Number Theory Call for Papers Number theory (arithmetic) is devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). In this special issue, we intend to invite front-line researchers and authors to submit original research and review articles on number theory. Potential topics include, but are not limited to: Elementary tools Analytic number theory Algebraic number theory Diophantine geometry Probabilistic number theory Arithmetic combinatorics Computational number theory Applications Authors should read over the journal’s For Authors carefully before submission. Prospective authors should submit an electronic copy of their complete manuscript through the journal’s Paper Submission System. Please kindly notice that the “Special Issue” under your manuscript title is supposed to be specified and the research field “Special Issue – Number Theory” should be chosen during your submission. According to the following timetable: Submission Deadline November 2nd, 2020 Publication Date January 2021 For publishing inquiries, please feel free to contact the Editorial Assistant at [email protected] APM Editorial Office Home | About SCIRP | Sitemap | Contact Us Copyright © 2006-2020 Scientific Research Publishing Inc. All rights reserved. Advances in Pure Mathematics Scientific Research Open Access ISSN Online: 2160-0384 [email protected] Home | About SCIRP | Sitemap | Contact Us Copyright © 2006-2020 Scientific Research Publishing Inc.
  • Ergodic Theory and Combinatorial Number Theory

    Ergodic Theory and Combinatorial Number Theory

    Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2012 DOI: 10.4171/OWR/2012/50 Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory Organised by Vitaly Bergelson, Columbus Nikos Frantzikinakis, Heraklion Terence Tao, Los Angeles Tamar Ziegler, Haifa 7th October – 13th October 2012 Abstract. The aim of this Arbeitsgemeinschaft was to introduce young re- searchers with various backgrounds to the multifaceted and mutually perpet- uating connections between ergodic theory, topological dynamics, combina- torics, and number theory. Mathematics Subject Classification (2000): 05D10, 11K06, 11B25, 11B30, 27A45, 28D05, 28D15, 37A30, 37A45, 37B05. Introduction by the Organisers In 1977 Furstenberg gave an ergodic proof of the celebrated theorem of Sze- mer´edi on arithmetic progressions, hereby starting a new field, ergodic Ramsey theory. Over the years the methods of ergodic theory and topological dynamics have led to very impressive developments in the fields of arithmetic combinatorics and Ramsey theory. Furstenberg’s original approach has been enhanced with sev- eral deep structural results of measure preserving systems, equidistribution results on nilmanifolds, the use of ultrafilters etc. Several novel techniques have been de- veloped and opened new vistas that led to new deep results, including far reaching extensions of Szemer´edi’s theorem, results in Euclidean Ramsey theory, multiple recurrence results for non-abelian group actions, convergence results for multiple ergodic averages etc. These methods have also facilitated the recent spectacu- lar progress on patterns in primes. The field of ergodic theory has tremendously benefited, since the problems of combinatorial and number-theoretic nature have given a boost to the in depth study of recurrence and convergence problems.
  • Contemporary Mathematics 300

    Contemporary Mathematics 300

    CONTEMPORARY MATHEMATICS 300 Algebraic Number Theory and Algebraic Geometry Papers Dedicated to A. N. Parshin on the Occasion of his Sixtieth Birthday Sergei Vostokov Yuri Zorhin Editors http://dx.doi.org/10.1090/conm/300 Algebraic Number Theory and Algebraic Geometry Aleksey Nikolaevich Parshin CoNTEMPORARY MATHEMATICS 300 Algebraic Number Theory and Algebraic Geometry Papers Dedicated to A. N. Parshin on the Occasion of his Sixtieth Birthday Sergei Vostokov Yuri Zarhin Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius 2000 Mathematics Subject Classification. Primary 11815, 11831, 14E22, 14F20, 14H30, 14H40, 14K10, 14K99, 14L05. Library of Congress Cataloging-in-Publication Data Algebraic number theory and algebraic geometry : papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday / Sergei Vostokov, Yuri Zarhin, editors. p. em. -(Contemporary mathematics; ISSN 0271-4132; v. 300) Includes bibliographical references. ISBN 0-8218-3267-0 (softcover : alk. paper) 1. Algebraic number theory. 2. Geometry, algebraic. I. Parshin, A. N. II. Vostokov, S. V. III. Zarhin, Yuri, 1951- IV. Contemporary mathematics (American Mathematical Society) ; v. 300. QA247 .A52287 2002 5121.74--dc21 2002074698 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
  • FROM HARMONIC ANALYSIS to ARITHMETIC COMBINATORICS: a BRIEF SURVEY the Purpose of This Note Is to Showcase a Certain Line Of

    FROM HARMONIC ANALYSIS to ARITHMETIC COMBINATORICS: a BRIEF SURVEY the Purpose of This Note Is to Showcase a Certain Line Of

    FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS: A BRIEF SURVEY IZABELLA ÃLABA The purpose of this note is to showcase a certain line of research that connects harmonic analysis, speci¯cally restriction theory, to other areas of mathematics such as PDE, geometric measure theory, combinatorics, and number theory. There are many excellent in-depth presentations of the vari- ous areas of research that we will discuss; see e.g., the references below. The emphasis here will be on highlighting the connections between these areas. Our starting point will be restriction theory in harmonic analysis on Eu- clidean spaces. The main theme of restriction theory, in this context, is the connection between the decay at in¯nity of the Fourier transforms of singu- lar measures and the geometric properties of their support, including (but not necessarily limited to) curvature and dimensionality. For example, the Fourier transform of a measure supported on a hypersurface in Rd need not, in general, belong to any Lp with p < 1, but there are positive results if the hypersurface in question is curved. A classic example is the restriction theory for the sphere, where a conjecture due to E. M. Stein asserts that the Fourier transform maps L1(Sd¡1) to Lq(Rd) for all q > 2d=(d¡1). This has been proved in dimension 2 (Fe®erman-Stein, 1970), but remains open oth- erwise, despite the impressive and often groundbreaking work of Bourgain, Wol®, Tao, Christ, and others. We recommend [8] for a thorough survey of restriction theory for the sphere and other curved hypersurfaces. Restriction-type estimates have been immensely useful in PDE theory; in fact, much of the interest in the subject stems from PDE applications.
  • Green-Tao Theorem in Function Fields 11

    Green-Tao Theorem in Function Fields 11

    GREEN-TAO THEOREM IN FUNCTION FIELDS THAI´ HOANG` LE^ Abstract. We adapt the proof of the Green-Tao theorem on arithmetic progressions in primes to the setting of polynomials over a finite fields, to show that for every k, the irreducible polynomials in Fq[t] contains configurations of the form ff + P g : deg(P ) < kg; g 6= 0. 1. Introduction In [13], Green and Tao proved the following celebrated theorem now bearing their name: Theorem 1 (Green-Tao). The primes contain arithmetic progressions of arbitrarily length. Furthermore, the same conclusion is true for any subset of positive relative upper density of the primes. Subsequently, other variants of this theorem have been proved. Tao and Ziegler [26] proved the generalization for polynomial progressions a + p1(d); : : : ; a + pk(d), where pi 2 Z[x] and pi(0) = 0. Tao [24] proved the analog in the Gaussian integers. It is well known that the integers and the polynomials over a finite field share a lot of similarities in many aspects relevant to arithmetic combinatorics. Therefore, it is natural, as Green and Tao did, to suggest that the analog of this theorem should hold in the setting of function fields: Conjecture 1. For any finite field F, the monic irreducible polynomials in F[t] contain affine spaces of arbitrarily high dimension. We give an affirmative answer to this conjecture. More precisely, we will prove: Theorem 2 (Green-Tao for function fields). Let Fq be a finite field over q elements. Then for any k > 0, we can find polynomials f; g 2 Fq[t]; g 6= 0 such that the polynomials f + P g, where P runs over all polynomials P 2 Fq[t] of degree less than k, are all irreducible.
  • PROBLEMS in ARITHMETIC TOPOLOGY Three Problem

    PROBLEMS in ARITHMETIC TOPOLOGY Three Problem

    PROBLEMS IN ARITHMETIC TOPOLOGY CLAUDIO GÓMEZ-GONZÁLES AND JESSE WOLFSON Abstract. We present a list of problems in arithmetic topology posed at the June 2019 PIMS/NSF workshop on "Arithmetic Topology". Three problem sessions were hosted during the workshop in which participants proposed open questions to the audience and engaged in shared discussions from their own perspectives as working mathematicians across various fields of study. Partic- ipants were explicitly asked to provide problems of various levels of difficulty, with the goal of capturing a cross-section of exciting challenges in the field that could help guide future activity. The problems, together with references and brief discussions when appropriate, are collected below into three categories: 1) topological analogues of arithmetic phenomena, 2) point counts, stability phenomena and the Grothendieck ring, and 3) tools, methods and examples. Three problem sessions were hosted during the workshop in which participants proposed open questions to the audience and engaged in shared discussions from their own perspectives as working mathematicians across various fields of study. Participants were explicitly asked to provide problems of various levels of difficulty, with the goal of capturing a cross-section of exciting challenges in the field that could help guide future activity. The problems, together with references and brief discussions when appropriate, are collected below into three categories: 1) topolog- ical analogues of arithmetic phenomena, 2) point counts, stability phenomena and the Grothendieck ring, and 3) tools, methods and examples. Acknowledgements. We thank all of the people who posed problems, both for their original suggestions and for their help clarifying and refining our write-ups of them.
  • The Cauchy Davenport Theorem

    The Cauchy Davenport Theorem

    The Cauchy Davenport Theorem Arithmetic Combinatorics (Fall 2016) Rutgers University Swastik Kopparty Last modified: Thursday 22nd September, 2016 1 Introduction and class logistics • See http://www.math.rutgers.edu/~sk1233/courses/additive-F16/ for the course website (including syllabus). • Class this Thursday is cancelled. We will schedule a makeup class sometime. • Office hours: Thursdays at 11am. • References: Tao and Vu, Additive combinatorics, and other online references (including class notes). • Grading: there will be 2 or 3 problem sets. 2 How small can a sumset be? Let A; B be subsets of an abelian group (G; +). The sumset of A and B, denoted A + B is given by: A + B = fa + b j a 2 A; b 2 Bg: We will be very interested in how the size of A + B relates to the sizes of A and B (for A,B finite). Some general comments. If A and B are generic, then we expect jA + Bj to be big. Only when A and B are very additively structured, and furthermore if their structure is highly compatible, does jA + Bj end up small. If G is the group of real numbers under addition, then the following simple inequality holds: jA + Bj ≥ jAj + jBj − 1: Proof: Let A = fa1; : : : ; akg where a1 < a2 < : : : < ak. Let B = fb1; : : : ; b`g, where b1 < : : : < b`. Then a1 + b1 < a1 + b2 < : : : < a1 + b` < a2 + b` < a3 + b` < : : : < ak + b`, and thus all these elements of A + B are distinct. Thus we found jAj + jBj − 1 distinct elements in A + B. Equality is attained if and only if A and B are arithmetic progressions with the same common difference (In case of equality we need to have either ai + bj = a1 + bj+i−1 or ai + bj = ai+j−` + b`).
  • Limits of Discrete Structures

    Limits of Discrete Structures

    Limits of discrete structures Balázs Szegedy Rényi Institute, Budapest Supported by the ERC grant: Limits of discrete structures The case k = 3 was solved by Roth in 1953 using Fourier analysis. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω ! Ω be an invertible measure preserving transformation on a probability space (Ω; B; µ) and let S 2 B be a set of positive measure. Then for every k there is d > 0 and x 2 Ω such that x; T d x; T 2d x;:::; T (k−1)d x 2 S In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem. A far-reaching story in mathematics Erdős-Turán conjecture (1936): For every k 2 N and > 0 there is N such that in every subset S ⊆ f1; 2;:::; Ng with jSj=N ≥ there is a k-term arithmetic progression a; a + b; a + 2b;:::; a + (k − 1)b with b 6= 0 contained in S. Szemerédi solved the Erdős-Turán conjecture in 1974. 1976: Furstenberg found an analytic approach using measure preser- ving systems. Furstenberg multiple recurrence: Let T :Ω ! Ω be an invertible measure preserving transformation on a probability space (Ω; B; µ) and let S 2 B be a set of positive measure. Then for every k there is d > 0 and x 2 Ω such that x; T d x; T 2d x;:::; T (k−1)d x 2 S In 1998 Gowers found an extension of Fourier analysis (called higher order Fourier analysis) that can be used to give explicit bounds in Szemerédi’s theorem.
  • Fundamental Groups of Number Fields Farshid Hajir in This Mostly Expository Lecture Aimed at Low-Dimensional Topologists, I

    Fundamental Groups of Number Fields Farshid Hajir in This Mostly Expository Lecture Aimed at Low-Dimensional Topologists, I

    Fundamental groups of number fields Farshid Hajir In this mostly expository lecture aimed at low-dimensional topologists, I out- lined some basic facts and problems of algebraic number theory. My focus was on one particular aspect of the rich set of analogies between number fields and 3-manifolds dubbed Arithmetic Topology. Namely, I discussed the role played in number theory by \fundamental groups" of number fields, and related some of the history of the subject over the past fifty years, since the unexpected discovery by Golod and Shafarevich of number fields with infinite fundamental group; see the monograph of Neukirch, Schmidt, Wingberg [10] for a comprehensive account. A conjecture of Fontaine and Mazur [3] has been influential in stimulating work on the structure of these infinite fundamental groups in recent years. I presented a for- mulation of this conjecture as it relates to the asymptotic growth of discriminants [6]. This discussion then served as motivation for a question about non-compact, finite-volume, 3-manifolds inspired by the following dictionary. Topology Arithmetic M non-compact, finite-volume K a number field hyperbolic 3-manifold or, more precisely, X = SpecOK universal cover Mf Ke = max. unramified extension of K et fundamental group π1(M) Gal(K=Ke ) ≈ π1 (X) Klein-bottle cusps of M Real (\unoriented") places of K Torus cusps of M Complex (\oriented") places of K r1 = # Klein-bottle cusps of M r1 = # Real places of K r2 = # Torus cusps of M r2 = # Complex places of K r = r1 + r2 = # cusps of M r = r1 + r2 = # places of K at 1 n = r1 + 2r2 = weighted # cusps n = r1 + 2r2 = [K : Q] vol(M) = volume of M log jdK j, dK = discriminant of K There are multiple accounts of the dictionary of arithmetic topology; these in- clude Reznikov [12], Ramachandran [11], Deninger [2], Morin [9], and Morishita [8].
  • Knots in Number Theory

    Knots in Number Theory

    T.D. van Mulligen Knots in Number Theory Master thesis, September 2017 Thesis supervisor: Dr. R.I. van der Veen Second supervisor: Dr. G.S. Zalamansky Mathematisch Instituut, Universiteit Leiden Contents 1. Introduction ............................................1 2. Topological preliminaries .....................................5 2.1 Knots............................................5 2.2 Covering spaces......................................7 3. Algebraic preliminaries ...................................... 13 3.1 Profinite groups...................................... 13 3.2 Affine schemes...................................... 15 3.3 Finite étale coverings................................... 17 Interlude on Galois categories................................. 19 3.4 Étale fundamental groups................................ 20 4. The linking number and the Legendre symbol .......................... 23 4.1 The linking number.................................... 23 4.2 The mod 2 linking number for primes......................... 26 5. Decomposition of knots and primes ................................ 31 5.1 Decomposition of knots................................. 31 5.2 Decomposition of primes................................. 37 6. Homology groups and ideal class groups ............................. 43 6.1 Homology groups and the Hurewicz theorem.................... 43 6.2 Ideal class groups and Artin reciprocity........................ 44 ii Contents 7. The Alexander and Iwasawa polynomials ............................. 47 7.1 Differential modules..................................
  • Past Programs I～VIII

    Past Programs I～VIII

    Low dimensional topology and number theory March 17-20, 2009 Soft Research Park Center, Fukuoka Program March 17 11:00 { 12:00 Susan Williams (University of South Alabama) Twisted Alexander invariants and two-bridge knots 13:30 { 14:30 Yuichi Kabaya (Tokyo Institute of Technology) A method for finding ideal points from an ideal triangulation and its appli- cation 14:50 { 15:50 Stavros Garoufalidis (Georgia Institute of Technology) Asymptotics of classical spin networks March 18 10:00 { 11:00 Daniel Silver (University of South Alabama) On a theorem of Burde and de Rham 11:20 { 12:20 Takahiro Kitayama (The University of Tokyo) Torsion volume forms and twisted Alexander functions on character varieties of knots 14:00 { 15:00 Kazuo Habiro (RIMS, Kyoto University) On the Witten-Reshetikhin-Turaev invariant and analytic functions on roots of unity 1 15:20 { 16:20 Don Zagier (Max Planck Institute) Modular properties of topological invariants and other q-series March 19 10:00 { 11:00 Jonathan Hillman (University of Sydney) Indecomposable PD3-complexes 11:20 { 12:20 Baptiste Morin (University of Bordeaux) On the Weil-´etalefundamental group of a number field 14:00 { 15:00 Ken-ichi Sugiyama (Chiba University) On a geometric analog of Iwasawa conjecture 15:00 { 15:25 Walter Neumann (Columbia University) Universal abelian covers in geometry and number theory March 20 10:00 { 11:00 Thang Le (Georgia Institute of Technology) Hyperbolic volumes, Mahler measure and homology growth 11:20 { 12:20 Shinya Harada (Kyushu University) Hasse-Weil zeta function