Geometry of Numbers with Applications to Number Theory
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Minkowski's Geometry of Numbers
Geometry of numbers MAT4250 — Høst 2013 Minkowski’s geometry of numbers Preliminary version. Version +2✏ — 31. oktober 2013 klokken 12:04 1 Lattices Let V be a real vector space of dimension n.Byalattice L in V we mean a discrete, additive subgroup. We say that L is a full lattice if it spans V .Ofcourse,anylattice is a full lattice in the subspace it spans. The lattice L is discrete in the induced topology, meaning that for any point x L 2 there is an open subset U of V whose intersection with L is just x.Forthesubgroup L to be discrete it is sufficient that this holds for the origin, i.e., that there is an open neigbourhood U about 0 with U L = 0 .Indeed,ifx L,thenthetranslate \ { } 2 U 0 = U + x is open and intersects L in the set x . { } Aoftenusefulpropertyofalatticeisthatithasonlyfinitelymanypointsinany bounded subset of V .Toseethis,letS V be a bounded set and assume that L S is ✓ \ infinite. One may then pick a sequence xn of different elements from L S.SinceS { } \ is bounded, its closure is compact and xn has a subsequence that converges, say to { } x. Then x is an accumulation point in L;everyneigbourhoodcontainselementsfrom L distinct from x. The lattice associated to a basis If = v1,...,vn is any basis for V ,onemayconsiderthesubgroupL = Zvi of B { } B i V consisting of all linear combinations of the vi’s with integral coefficients. Clearly L P B is a discrete subspace of V ;IfonetakesU to be an open ball centered at the origin and having radius less than mini vi ,thenU L = 0 .HenceL is a full lattice in k k \ { } B V . -
Chapter 2 Geometry of Numbers
Chapter 2 Geometry of numbers Literature: W.M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics 785, Springer Verlag 1980, Chap.II, xx1,2, Chap. IV, x1 J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer Verlag 1997, Classics in Mathematics series, reprint of the 1971 edition C.L. Siegel, Lectures on the Geometry of Numbers, Springer Verlag 1989 2.1 Introduction Geometry of numbers is concerned with the study of lattice points in certain bodies n in R , where n > 2. We discuss Minkowski's theorems on lattice points in central symmetric convex bodies. In this introduction we give the necessary definitions. Lattices. A (full) lattice in Rn is an additive group L = fz1v1 + ··· + znvn : z1; : : : ; zn 2 Zg n where fv1;:::; vng is a basis of R , i.e., fv1;:::; vng is linearly independent. We call fv1;:::; vng a basis of L. The determinant of L is defined by d(L) := j det(v1;:::; vn)j; that is, the absolute value of the determinant of the matrix with columns v1;:::; vn. 7 We show that the determinant of a lattice does not depend on the choice of the basis. Recall that GL(n; Z) is the multiplicative group of n×n-matrices with entries in Z and determinant ±1. Lemma 2.1. Let L be a lattice, and fv1;:::; vng, fw1;:::; wng two bases of L. Then there is a matrix U = (uij) 2 GL(n; Z) such that n X (2.1) wi = uijvj for i = 1; : : : ; n: j=1 Consequently, j det(v1;:::; vn)j = j det(w1;:::; wn)j. -
Introductory Number Theory Course No
Introductory Number Theory Course No. 100 331 Spring 2006 Michael Stoll Contents 1. Very Basic Remarks 2 2. Divisibility 2 3. The Euclidean Algorithm 2 4. Prime Numbers and Unique Factorization 4 5. Congruences 5 6. Coprime Integers and Multiplicative Inverses 6 7. The Chinese Remainder Theorem 9 8. Fermat’s and Euler’s Theorems 10 × n × 9. Structure of Fp and (Z/p Z) 12 10. The RSA Cryptosystem 13 11. Discrete Logarithms 15 12. Quadratic Residues 17 13. Quadratic Reciprocity 18 14. Another Proof of Quadratic Reciprocity 23 15. Sums of Squares 24 16. Geometry of Numbers 27 17. Ternary Quadratic Forms 30 18. Legendre’s Theorem 32 19. p-adic Numbers 35 20. The Hilbert Norm Residue Symbol 40 21. Pell’s Equation and Continued Fractions 43 22. Elliptic Curves 50 23. Primes in arithmetic progressions 66 24. The Prime Number Theorem 75 References 80 2 1. Very Basic Remarks The following properties of the integers Z are fundamental. (1) Z is an integral domain (i.e., a commutative ring such that ab = 0 implies a = 0 or b = 0). (2) Z≥0 is well-ordered: every nonempty set of nonnegative integers has a smallest element. (3) Z satisfies the Archimedean Principle: if n > 0, then for every m ∈ Z, there is k ∈ Z such that kn > m. 2. Divisibility 2.1. Definition. Let a, b be integers. We say that “a divides b”, written a | b , if there is an integer c such that b = ac. In this case, we also say that “a is a divisor of b” or that “b is a multiple of a”. -
Exercises for Geometry of Numbers Week 8: 01.05.2020
EXERCISES FOR GEOMETRY OF NUMBERS WEEK 8: 01.05.2020 Exercise 1. 1. Consider all the intervals of the form [a2t; (a + 1)2t] with a; t non- ∗ negative integers. For s 2 N , let Ls be the collection of such intervals with t ≤ s − 1 s s contained in [0; 2 ]. Show that for any k 2 N with k ≤ 2 , the interval [0; k] can be covered by at most s elements of Ls. 2. Let F : [0; 1] × (0; 1) be a bounded measurable function and suppose that there exists c > 0 such that for every interval (a; b) ⊂ (0; 1), we have Z 1 Z b 2 F (x; t)dt dx ≤ c(b − a): 0 a Show that 2 X Z 1 Z F (x; t)dt dx ≤ cs2s: 0 I I2Ls ∗ 3. Fix > 0. Show that for every s 2 N , there exists a measurable set As ⊂ [0; 1] such −1−" 1 s that jAsj = O(s ) and for every y2 = As and for every k 2 N with k ≤ 2 , we have Z k s 3 + j F (t; y)dtj ≤ 2 2 s 2 : 0 4. Deduce from 3. 2 that for a.e. y 2 [0; 1], we have Z T 1 − 1 3 j F (y; t)dtj = O(T 2 log(T ) 2 ) T 0 Although the arithmetic functions are not the main focus of the course, the next two exercises touch upon those and aim at proving the arithmetic estimate used in Schmidt's a.s. counting theorem. -
Artin L-Functions and Arithmetic Equivalence (Evan Dummit, September 2013)
Artin L-Functions and Arithmetic Equivalence (Evan Dummit, September 2013) 1 Intro • This is a prep talk for Guillermo Mantilla-Soler's talk. • There are approximately 3 parts of this talk: ◦ First, I will talk about Artin L-functions (with some examples you should know) and in particular try to explain very vaguely what local root numbers are. This portion is adapted from Neukirch and Rohrlich. ◦ Second, I will do a bit of geometry of numbers and talk about quadratic forms and lattices. ◦ Finally, I will talk about arithmetic equivalence and try to give some of the broader context for Guillermo's results (adapted mostly from his preprints). • Just to give the avor of things, here is the theorem Guillermo will probably be talking about: ◦ Theorem (Mantilla-Soler): Let K; L be two non-totally-real, tamely ramied number elds of the same discriminant and signature. Then the integral trace forms of K and L are isometric if and only if for all odd primes p dividing disc(K) the p-local root numbers of ρK and ρL coincide. 2 Artin L-Functions and Root Numbers • Let L=K be a Galois extension of number elds with Galois group G, and ρ be a complex representation of G, which we think of as ρ : G ! GL(V ). • Let p be a prime of K, Pjp a prime of L above p, with kL = OL=P and kK = OK =p the corresponding residue elds, and also let GP and IP be the decomposition and inertia groups of P above p. ∼ • By standard things, the group GP=IP = G(kL=kK ) is generated by the Frobenius element FrobP (which in the group on the right is just the standard q-power Frobenius where q = Nm(p)), so we can think of it as acting on the invariant space V IP . -
Symmetric Coverage of Dynamic Mapping Error for Mobile Sensor Networks
2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 Symmetric coverage of dynamic mapping error for mobile sensor networks Carlos H. Caicedo-Nu´nez˜ and Naomi Ehrich Leonard Abstract—We present an approach to control design for nonuniform density fields. Related problems include control a mobile sensor network tasked with sampling a scalar field of a mobile sensor network in a noisy, possibly time-varying and providing optimal space-time measurements. The coverage field for minimum-error spatial estimation [9], [10] and metric is derived from the mapping error in objective analysis (OA), an assimilation scheme that provides a linear statistical cooperative exploration of features [11]. In [12] agents are estimation of a sampled field. OA mapping error is an example deployed in a distributed way in order to maximize the of a consumable density field: the error decreases dynamically probability of event detection. The authors of [13] study the at locations where agents move and sample. OA mapping error distributed implementation of maximizing joint entropy in is also a regenerating density field if the sampled field is measurements. However, none of these methods have been time-varying: error increases over time as measurement value decays. The resulting optimal coverage problem presents a chal- designed to specifically address a spatial-temporal density lenge to traditional coverage methods. We prove a symmetric field that changes in response to the motion of the agents. dynamic coverage solution that exploits the symmetry of the In this paper we propose a coverage strategy for a density domain and yields symmetry-preserving coordinated motion field defined by OA mapping error in the case that the of mobile sensors. -
On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017
On Some Variants of the Gauss Circle Problem by David Lowry-Duda B.S. in Applied Mathematics, Georgia Institute of Technology, Atlanta, GA, 2011 B.S. in International Affairs and Modern Languages, Georgia Institute of Technology, Atlanta, GA, 2011 M.Sc. in Mathematics, Brown University, Providence, RI, 2015 A dissertation submitted in partial fulfillment of the arXiv:1704.02376v2 [math.NT] 2 May 2017 requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2017 c Copyright 2017 by David Lowry-Duda Abstract of “On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017 The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly accurate. This seemingly simple problem has prompted new ideas in many areas of number theory and mathematics, and it is now recognized as one instance of a general phenomenon. In this work, we describe two variants of the Gauss Circle problem that exhibit similar characteristics. The first variant concerns sums of Fourier coefficients of GL(2) cusp forms. These sums behave very similarly to the error term in the Gauss Circle problem. Normalized correctly, it is conjectured that the two satisfy essentially the same asymptotics. We introduce new Dirichlet series with coefficients that are squares of partial sums of Fourier coefficients of cusp forms. We study the meromorphic properties of these Dirichlet series and use these series to give new perspectives on the mean square of the size of sums of these Fourier coefficients. -
Journal of Computational and Applied Mathematics Numerical Integration
Journal of Computational and Applied Mathematics 233 (2009) 279–292 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Numerical integration over polygons using an eight-node quadrilateral spline finite elementI Chong-Jun Li a,∗, Paola Lamberti b, Catterina Dagnino b a School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China b Department of Mathematics, University of Torino, via C. Alberto, 10 Torino 10123, Italy article info a b s t r a c t Article history: In this paper, a cubature formula over polygons is proposed and analysed. It is based Received 9 February 2009 on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node Received in revised form 8 July 2009 quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on MSC: rectangular partitions. The convergence of sequences of the above cubatures is proved for 65D05 continuous integrand functions and error bounds are derived. Some numerical examples 65D07 are given, by comparisons with other known cubatures. 65D30 65D32 ' 2009 Elsevier B.V. All rights reserved. Keywords: Numerical integration Spline finite element method Bivariate splines Triangulated quadrangulation 1. Introduction The problem considered in this paper is the numerical evaluation of Z IΩ .f / D f .x; y/dxdy; (1) Ω 2 where f 2 C.Ω/ and Ω is a polygonal domain in R , i.e. a domain with the boundary composed of piecewise straight lines. -
MATHEMATICAL TRIPOS Part IB 2017
MATHEMATICALTRIPOS PartIB 2017 List of Courses Analysis II Complex Analysis Complex Analysis or Complex Methods Complex Methods Electromagnetism Fluid Dynamics Geometry Groups, Rings and Modules Linear Algebra Markov Chains Methods Metric and Topological Spaces Numerical Analysis Optimisation Quantum Mechanics Statistics Variational Principles Part IB, 2017 List of Questions [TURN OVER 2 Paper 3, Section I 2G Analysis II What does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers. (i) [0, 1] with the Euclidean metric. (ii) Q with the Euclidean metric. (iii) The subset (0, 0) (x, sin(1/x)) x> 0 R2 { } ∪ { | }⊂ with the metric induced from the Euclidean metric on R2. Write down a metric on R with respect to which R is not complete, justifying your answer. [You may assume throughout that R is complete with respect to the Euclidean metric.] Paper 2, Section I 3G Analysis II Let X R. What does it mean to say that a sequence of real-valued functions on ⊂ X is uniformly convergent? Let f,f (n > 1): R R be functions. n → (a) Show that if each fn is continuous, and (fn) converges uniformly on R to f, then f is also continuous. (b) Suppose that, for every M > 0, (f ) converges uniformly on [ M, M]. Need n − (fn) converge uniformly on R? Justify your answer. Paper 4, Section I 3G Analysis II State the chain rule for the composition of two differentiable functions f : Rm Rn → and g : Rn Rp. → Let f : R2 R be differentiable. -
Lattices and the Geometry of Numbers
Lattices and the Geometry of Numbers Lattices and the Geometry of Numbers Sourangshu Ghosha, aUndergraduate Student ,Department of Civil Engineering , Indian Institute of Technology Kharagpur, West Bengal, India 1. ABSTRACT In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to study the properties of algebraic numbers. It has application on various other fields of mathematics especially the study of Diophantine equations, analysis of functional analysis etc. This paper will review all the major developments that have occurred in the field of geometry of numbers. In this paper we shall first give a broad overview of the concept of lattice and then discuss about the geometrical properties it has and its applications. 2. LATTICE Before introducing Minkowski’s theorem we shall first discuss what is a lattice. Definition 1: A lattice 흉 is a subgroup of 푹풏 such that it can be represented as 휏 = 푎1풁 + 푎2풁 + . + 푎푚풁 풏 Here {푎푖} are linearly independent vectors of the space 푹 and 푚 ≤ 푛. Here 풁 is the set of whole numbers. We call these vectors {푎푖} the basis of the lattice. By the definition we can see that a lattice is a subgroup and a free abelian group of rank m, of the vector space 푹풏. The rank and dimension of the lattice is 푚 and 푛 respectively, the lattice will be complete if 푚 = 푛. This definition is not only limited to the vector space 푹풏. -
Arithmetic of L-Functions
ARITHMETIC OF L-FUNCTIONS CHAO LI NOTES TAKEN BY PAK-HIN LEE Abstract. These are notes I took for Chao Li's course on the arithmetic of L-functions offered at Columbia University in Fall 2018 (MATH GR8674: Topics in Number Theory). WARNING: I am unable to commit to editing these notes outside of lecture time, so they are likely riddled with mistakes and poorly formatted. Contents 1. Lecture 1 (September 10, 2018) 4 1.1. Motivation: arithmetic of elliptic curves 4 1.2. Understanding ralg(E): BSD conjecture 4 1.3. Bridge: Selmer groups 6 1.4. Bloch{Kato conjecture for Rankin{Selberg motives 7 2. Lecture 2 (September 12, 2018) 7 2.1. L-functions of elliptic curves 7 2.2. L-functions of modular forms 9 2.3. Proofs of analytic continuation and functional equation 10 3. Lecture 3 (September 17, 2018) 11 3.1. Rank part of BSD 11 3.2. Computing the leading term L(r)(E; 1) 12 3.3. Rank zero: What is L(E; 1)? 14 4. Lecture 4 (September 19, 2018) 15 L(E;1) 4.1. Rationality of Ω(E) 15 L(E;1) 4.2. What is Ω(E) ? 17 4.3. Full BSD conjecture 18 5. Lecture 5 (September 24, 2018) 19 5.1. Full BSD conjecture 19 5.2. Tate{Shafarevich group 20 5.3. Tamagawa numbers 21 6. Lecture 6 (September 26, 2018) 22 6.1. Bloch's reformulation of BSD formula 22 6.2. p-Selmer groups 23 7. Lecture 7 (October 8, 2018) 25 7.1. -
Complex Analysis
University of Cambridge Mathematics Tripos Part IB Complex Analysis Lent, 2018 Lectures by G. P. Paternain Notes by Qiangru Kuang Contents Contents 1 Basic notions 2 1.1 Complex differentiation ....................... 2 1.2 Power series .............................. 4 1.3 Conformal maps ........................... 8 2 Complex Integration I 10 2.1 Integration along curves ....................... 10 2.2 Cauchy’s Theorem, weak version .................. 15 2.3 Cauchy Integral Formula, weak version ............... 17 2.4 Application of Cauchy Integration Formula ............ 18 2.5 Uniform limits of holomorphic functions .............. 20 2.6 Zeros of holomorphic functions ................... 22 2.7 Analytic continuation ........................ 22 3 Complex Integration II 24 3.1 Winding number ........................... 24 3.2 General form of Cauchy’s theorem ................. 26 4 Laurent expansion, Singularities and the Residue theorem 29 4.1 Laurent expansion .......................... 29 4.2 Isolated singularities ......................... 30 4.3 Application and techniques of integration ............. 34 5 The Argument principle, Local degree, Open mapping theorem & Rouché’s theorem 38 Index 41 1 1 Basic notions 1 Basic notions Some preliminary notations/definitions: Notation. • 퐷(푎, 푟) is the open disc of radius 푟 > 0 and centred at 푎 ∈ C. • 푈 ⊆ C is open if for any 푎 ∈ 푈, there exists 휀 > 0 such that 퐷(푎, 휀) ⊆ 푈. • A curve is a continuous map from a closed interval 휑 ∶ [푎, 푏] → C. It is continuously differentiable, i.e. 퐶1, if 휑′ exists and is continuous on [푎, 푏]. • An open set 푈 ⊆ C is path-connected if for every 푧, 푤 ∈ 푈 there exists a curve 휑 ∶ [0, 1] → 푈 with endpoints 푧, 푤.