Geometric Number Theory Lenny Fukshansky Minkowski's creation of the geometry of numbers was likened to the story of Saul, who set out to look for his father's asses and discovered a Kingdom. J. V. Armitage Contents Chapter 1. Geometry of Numbers 1 1.1. Introduction 1 1.2. Norms, sets, and volumes 2 1.3. Lattices 7 1.4. Theorems of Blichfeldt and Minkowski 15 1.5. Successive minima 18 1.6. Inhomogeneous minimum 23 1.7. Problems 26 Chapter 2. Discrete Optimization Problems 30 2.1. Sphere packing, covering and kissing number problems 30 2.2. Lattice packings in dimension 2 36 2.3. Algorithmic problems on lattices 41 2.4. Problems 45 Chapter 3. Quadratic Forms 46 3.1. Introduction to quadratic forms 46 3.2. Minkowski's reduction 53 3.3. Sums of squares 56 3.4. Problems 60 Chapter 4. Diophantine Approximation 63 4.1. Real and rational numbers 63 4.2. Algebraic and transcendental numbers 65 4.3. Dirichlet's Theorem 69 4.4. Liouville's theorem and construction of a transcendental number 73 4.5. Roth's theorem 75 4.6. Continued fractions 78 4.7. Kronecker's theorem 84 4.8. Problems 87 Chapter 5. Algebraic Number Theory 90 5.1. Some field theory 90 5.2. Number fields and rings of integers 96 5.3. Noetherian rings and factorization 105 5.4. Norm, trace, discriminant 109 5.5. Fractional ideals 113 5.6. Further properties of ideals 117 5.7. Minkowski embedding 121 5.8. The class group 124 v vi CONTENTS 5.9. Dirichlet's unit theorem 127 5.10. Problems 131 Chapter 6. Transcendental Number Theory 135 6.1. Function fields and transcendence 135 6.2. Hermite, Lindemann, Weierstrass 138 6.3. Beyond Lindemann-Weierstrass 144 6.4. Siegel's Lemma 147 6.5. The Six Exponentials Theorem 151 6.6. Problems 154 Chapter 7. Further Topics 155 7.1. Frobenius problem 155 7.2. Lattice point counting in homogeneously expanding domains 162 7.3. Simultaneous Diophantine approximation 169 7.4. Absolute values and height functions 175 7.5. Mahler measure and Lehmer's problem 188 7.6. Points of small height 192 7.7. Problems 195 Appendices Appendix A. Some properties of abelian groups 198 Appendix B. Maximum Modulus Principle and Fundamental Theorem of Algebra 201 Appendix C. Brief remarks on exponential and logarithmic functions 203 Appendix. Bibliography 207 CHAPTER 1 Geometry of Numbers 1.1. Introduction The foundations of the Geometry of Numbers were laid down by Hermann Minkowski in his monograph \Geometrie der Zahlen", which was published in 1910, a year after his death. This subject is concerned with the interplay of compact convex 0-symmetric sets and lattices in Euclidean spaces. A set K ⊂ Rn is compact if it is closed and bounded, and it is convex if for any pair of points x; y 2 K the line segment connecting them is entirely contained in K, i.e. for any 0 ≤ t ≤ 1, tx + (1 − t)y 2 K. Further, K is called 0-symmetric if for any x 2 K, −x 2 K. Given such a set K in Rn, one can ask for an easy criterion to determine if K contains any nonzero points with integer coordinates. While for an arbitrary set K such a criterion can be rather difficult, in case of K as above a criterion purely in terms of its volume is provided by Minkowski's fundamental theorem. It is not difficult to see that K must in fact be convex and 0-symmetric for a criterion like this purely in terms of the volume of K to be possible. Indeed, the rectangle 2 R = (x; y) 2 R : 1=3 ≤ x ≤ 2=3; −t ≤ y ≤ t is convex for every t, but not 0-symmetric, and its area is 2t=3, which can be arbitrarily large depending on t while it still contains no integer points at all. On the other hand, the set R+ [ −R+ where R+ = f(x; y) 2 R : y ≥ 0g and −R+ = f(−x; −y):(x; y) 2 R+g is 0-symmetric, but not convex, and again can have arbitrarily large area while containing no integer points. Minkowski's theory applies not only to the integer lattice, but also to more general lattices. Our goal in this chapter is to introduce Minkowski's powerful theory, starting with the basic notions of lattices. 1 2 1. GEOMETRY OF NUMBERS 1.2. Norms, sets, and volumes Throughout this section we will work in the real vector space Rn, where n ≥ 1. Definition 1.2.1. A function F : Rn ! R is called a norm if (1) F (x) ≥ 0 with equality if and only if x = 0, (2) F (ax) = jajF (x) for each a 2 R, x 2 Rn, (3) Triangle inequality: F (x + y) ≤ F (x) + F (y) for all x; y 2 Rn. n For each positive integer p, we can introduce the Lp-norm k kp on R defined by n !1=p X p kxkp = jxij ; i=1 n for each x = (x1; : : : ; xn) 2 R . We also define the sup-norm, given by jxj = max jxij: 1≤i≤n These indeed are norms on Rn (Problem 1.1). Unless stated otherwise, we will regard Rn as a normed linear space (i.e. a vector space equipped with a norm) with respect to the Euclidean norm k k2: from now on we will refer to it simply as k k. Recall that for every two points x; y 2 Rn, Euclidean distance between them is given by d(x; y) = kx − yk: We start with definitions and examples of a few different types of subsets of Rn that we will often encounter. Definition 1.2.2. A subset X ⊆ Rn is called compact if it is closed and bounded. Recall that a set is closed if it contains all of its limit points, and it is bounded if there exists M 2 R>0 such that for every two points x; y in this set d(x; y) ≤ M. For instance, the closed unit ball centered at the origin in Rn n Bn = fx 2 R : kxk ≤ 1g is a compact set, but its interior, the open ball o n Bn = fx 2 R : kxk < 1g is not a compact set. If we now write n Sn−1 = fx 2 R : kxk = 1g n for the unit sphere centered at the origin in R , then it is easy to see that Bn = o Sn−1 [ Bn, and we refer to Sn−1 as the boundary of Bn (sometimes we will write o Sn−1 = @Bn) and to Bn as the interior of Bn. From here on we will also assume that all our compact sets have no isolated points. Then we can say more generally that every compact set X ⊂ Rn has boundary @X and interior Xo, and can be represented as X = @X [ Xo. To make this notation precise, we say that a point x 2 X is a boundary point of X if every open neighborhood U of x contains points in X and points not in X; we write @X for the set of all boundary points of X. All points x 2 X that are not in @X are called interior points of X, and we write Xo for the set of all interior points of X. 1.2. NORMS, SETS, AND VOLUMES 3 Definition 1.2.3. A compact subset X ⊆ Rn is called convex if whenever x; y 2 X, then any point of the form tx + (1 − t)y; where t 2 [0; 1], is also in X; i.e. whenever x; y 2 X, then the entire line segment from x to y lies in X. We now briefly mention a special class of convex sets. Given a set X in Rn, we define the convex hull of X to be the set ( ) X X Co(X) = txx : tx ≥ 0 8 x 2 X; tx = 1 : x2X x2X It is easy to notice that whenever a convex set contains X, it must also contain Co(X). Hence convex hull of a collection of points should be thought of as the smallest convex set containing all of them. If the set X is finite, then its convex hull is called a convex polytope. Most of the times we will be interested in convex polytopes, but occasionally we will also need convex hulls of infinite sets. There is an alternative way of describing convex polytopes. Recall that a hyperplane in Rn is a translate of a co-dimension one subspace, i.e. a subset H in Rn is called a hyperplane if ( n ) n X (1.1) H = x 2 R : aixi = b ; i=1 n for some a1; : : : ; an; b 2 R. Notice that each hyperplane divides R into two halfs- paces. More precisely, a closed halfspace H in Rn is a set of all x 2 Rn such that Pn Pn either i=1 aixi ≥ b or i=1 aixi ≤ b for some a1; : : : ; an; b 2 R. Minkowski-Weyl theorem (Problem 1.4) asserts that a set is a convex polytope in Rn if and only if it is a bounded intersection of finitely many halfspaces. Polytopes form a very nice class of convex sets in Rn, and we will talk more about them later. There is, of course, a large variety of sets that are not necessarily convex. Among these, ray sets and star bodies form a particularly nice class.