MA3H1 Topics in Number Theory Samir Siksek
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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 . -
Number Theory, Dover Publications, Inc., New York, 1994
Theory of Numbers Divisibility Theory in the Integers, The Theory of Congruences, Number-Theoretic Functions, Primitive Roots, Quadratic Residues Yotsanan Meemark Informal style based on the course 2301331 Theory of Numbers, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Second version August 2016 Any comment or suggestion, please write to [email protected] Contents 1 Divisibility Theory in the Integers 1 1.1 The Division Algorithm and GCD . 1 1.2 The Fundamental Theorem of Arithmetic . 5 1.3 The Euclidean Algorithm and Linear Diophantine Equations . 8 2 The Theory of Congruences 13 2.1 Basic Properties of Congruence . 13 2.2 Linear Congruences . 16 2.3 Reduced Residue Systems . 18 2.4 Polynomial Congruences . 21 3 Number-Theoretic Functions 25 3.1 Multiplicative Functions . 25 3.2 The Mobius¨ Inversion Formula . 28 3.3 The Greatest Integer Function . 30 4 Primitive Roots 33 4.1 The Order of an Integer Modulo n ............................ 33 4.2 Integers Having Primitive Roots . 35 4.3 nth power residues . 39 4.4 Hensel’s Lemma . 41 5 Quadratic Residues 45 5.1 The Legendre Symbol . 45 5.2 Quadratic Reciprocity . 48 Bibliography 53 Index 54 Chapter 1 Divisibility Theory in the Integers Let N denote the set of positive integers and let Z be the set of integers. 1.1 The Division Algorithm and GCD Theorem 1.1.1. [Well-Ordering Principle] Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a ≤ b for all b 2 S. -
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”. -
The Complete Splittings of Finite Abelian Groups
The complete splittings of finite abelian groups Kevin Zhaoa aDepartment of Mathematics, South China normal university, Guangzhou 510631, China Abstract Let G be a finite group. We will say that M and S form a complete split- ting (splitting) of G if every element (nonzero element) g of G has a unique representation of the form g = ms with m ∈ M and s ∈ S, and 0 has a such representation (while 0 has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our de- scription is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups. Keywords: complete splittings, splittings, cyclic groups, finite abelian groups, relationship. 1. Introduction The splittings of finite abelian groups are closely related to lattice tilings and are easy to be generated to lattice packings. Let K0 be a polytope composed of unit cubes and v + K0 be a translate of K0 for some vector v. A family of translations {v + K0 : v ∈ H} is called an integer lattice packing if H is an n-dimensional subgroup of Zn and, for any two vectors v and w in H, the interiors of v + K0 and w + K0 are disjoint; furthermore, if the n-dimensional Euclidean space Zn is contained in the family of these translations, then we call it an integer lattice tiling. The splitting problem can be traceable to a geometric problem posed by H. -
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 . -
Primes in Arithmetical Progression
Colby College Digital Commons @ Colby Honors Theses Student Research 2019 Primes in Arithmetical Progression Edward C. Wessel Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Analysis Commons, and the Number Theory Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Wessel, Edward C., "Primes in Arithmetical Progression" (2019). Honors Theses. Paper 935. https://digitalcommons.colby.edu/honorstheses/935 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. Primes in Arithmetical Progression Edward (Teddy) Wessel April 2019 Contents 1 Message to the Reader 3 2 Introduction 3 2.1 Primes . 3 2.2 Euclid . 3 2.3 Euler and Dirichlet . 4 2.4 Shapiro . 4 2.5 Formal Statement . 5 3 Arithmetical Functions 6 3.1 Euler’s Totient Function . 6 3.2 The Mobius Function . 6 3.3 A Relationship Between j and m ................ 7 3.4 The Mangoldt Function . 8 4 Dirichlet Convolution 9 4.1 Definition . 10 4.2 Some Basic Properties of Dirichlet Multiplication . 10 4.3 Identity and Inverses within Dirichlet Multiplication . 11 4.4 Multiplicative Functions and their Relationship with Dirich- let Convolution . 13 4.5 Generalized Convolutions . 15 4.6 Partial Sums of Dirichlet Convolution . -
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. -
Modular Arithmetic Some Basic Number Theory: CS70 Summer 2016 - Lecture 7A Midterm 2 Scores Out
Announcements Agenda Modular Arithmetic Some basic number theory: CS70 Summer 2016 - Lecture 7A Midterm 2 scores out. • Modular arithmetic Homework 7 is out. Longer, but due next Wednesday before class, • GCD, Euclidean algorithm, and not next Monday. multiplicative inverses David Dinh • Exponentiation in modular There will be no homework 8. 01 August 2016 arithmetic UC Berkeley Mathematics is the queen of the sciences and number theory is the queen of mathematics. -Gauss 1 2 Modular Arithmetic Motivation: Clock Math Congruences Modular Arithmetic If it is 1:00 now. What time is it in 2 hours? 3:00! Theorem: If a c (mod m) and b d (mod m), then a + b c + d x is congruent to y modulo m, denoted “x y (mod m)”... ≡ ≡ ≡ What time is it in 5 hours? 6:00! ≡ (mod m) and a b = c d (mod m). · · What time is it in 15 hours? 16:00! • if and only if (x y) is divisible by m (denoted m (x y)). Proof: Addition: (a + b) (c + d) = (a c) + (b d). Since a c − | − − − − ≡ Actually 4:00. • if and only if x and y have the same remainder w.r.t. m. (mod m) the first term is divisible by m, likewise for the second 16 is the “same as 4” with respect to a 12 hour clock system. • x = y + km for some integer k. term. Therefore the entire expression is divisible by m, so Clock time equivalent up to to addition/subtraction of 12. a + b c + d (mod m). What time is it in 100 hours? 101:00! or 5:00. -
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. -
Group Properties of the Residue Classes of Certain Kroneckermodular Systems and Some Related Generalizationsin Number Theory*
GROUP PROPERTIES OF THE RESIDUE CLASSES OF CERTAIN KRONECKERMODULAR SYSTEMS AND SOME RELATED GENERALIZATIONSIN NUMBER THEORY* BT EDWARD KIRCHER The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related gen- eralizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9JÎ = (mn, m„_i, • • •, mi, m). Here m, defined by ( mi, m2, ■• • , m* ), is an ideal in the algebraic domain Q of degree k. Each term m,, i = 1, 2, • • • , n, belongs to the domain of in- tegrity of Í2¿ = (ß, xi, x2, • • • , Xi), and is defined by the fundamental system ( ^'i0 ) typ, • - - > typ ) • The various typ, j = 1, 2, • • • , j,-, are rational integral functions of Xj with coefficients that are in turn rational integral func- tions of a:i, x2, • • •, Xi-i, with coefficients that are algebraic integers in ß. In every case the coefficient of the highest power of a-¿in each of the typ shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being re- placed by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each typ with the possible ex- ception of the last one, we shall call a polynomial, and no other expression shall be so designated.