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Geometry of Numbers with Applications to Number Theory

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GEOMETRY OF NUMBERS WITH APPLICATIONS TO NUMBER THEORY PETE L. CLARK Contents 1. Lattices in Euclidean Space 3 1.1. Discrete vector groups 3 1.2. Hermite and Smith Normal Forms 5 1.3. Fundamental regions, covolumes and sublattices 6 1.4. The Classification of Vector Groups 13 1.5. The space of all lattices 13 2. The Lattice Point Enumerator 14 2.1. Introduction 14 2.2. Gauss's Solution to the Gauss Circle Problem 15 2.3. A second solution to the Gauss Circle Problem 16 2.4. Introducing the Lattice Point Enumerator 16 2.5. Error Bounds on the Lattice Enumerator 18 2.6. When @Ω is smooth and positively curved 19 2.7. When @Ω is a fractal set 20 2.8. When Ω is a polytope 20 3. The Ehrhart (Quasi-)Polynomial 20 3.1. Basic Terminology 20 4. Convex Sets, Star Bodies and Distance Functions 21 4.1. Centers and central symmetry 21 4.2. Convex Subsets of Euclidean Space 22 4.3. Star Bodies 24 4.4. Distance Functions 24 4.5. Jordan measurability 26 5. Minkowski's Convex Body Theorem 26 5.1. Statement of Minkowski's First Theorem 26 5.2. Mordell's Proof of Minkowski's First Theorem 27 5.3. Statement of Blichfeldt's Lemma 27 5.4. Blichfeldt's Lemma Implies Minkowski's First Theorem 28 5.5. First Proof of Blichfeldt's Lemma: Riemann Integration 28 5.6. Second Proof of Blichfeldt's Lemma: Lebesgue Integration 28 5.7. A Strengthened Minkowski's First Theorem 29 5.8. Some Refinements 30 5.9. Pick's Theorem via Minkowski's Theorem 32 6. Minkowski's Theorem on Successive Minima 33 Thanks to Lauren Huckaba for a careful reading of x6, including spotting some typos and fleshing out some details in the proof of the Minkowski- Hlawka Theorem. 1 2 PETE L. CLARK 7. The Minkowski-Hlawka Theorem 34 7.1. Statement of the theorem 34 7.2. Proof of Minkowski-Hlawka, Part a) 35 7.3. Primitive Lattice Points 36 7.4. Proof of Minkowski-Hlawka Part b) 38 7.5. Proof of Minkowski-Hlawka Part c) 39 8. Mahler's Compactness Theorem 39 9. Lattice Points in Star Bodies 39 9.1. Lp norms 39 9.2. Linear Forms 40 9.3. Products of Linear Forms 41 9.4. Positive Definite Quadratic Forms 43 9.5. Binary Quadratic Forms 46 9.6. The Lattice Constant of a Star Body 50 10. More on Hermite Constants 51 10.1. Hermite's bound on the Hermite constant 51 10.2. The Known Hermite Constants 53 10.3. Mordell's Inequality 54 10.4. Computation of γ3 and γ4 55 10.5. Computation of γ2;1 and γ2;2 58 11. Applications of GoN: Algebraic Number Theory 58 11.1. Basic Setup 58 11.2. The Lattice Associated to an Ideal 59 11.3. A Standard Volume Calculation 59 11.4. Finiteness of the Class Group 60 11.5. Non-maximal orders 61 11.6. Other Finiteness Theorems 62 11.7. The Dirichlet Unit Theorem 63 11.8. The Lattice Associated to an S-Integer Ring 65 12. Applications of GoN: Linear Forms 66 12.1. Vinogradov's Lemma 66 12.2. Improvements on Vinogradov: Brauer-Reynolds and Cochrane 68 12.3. A Number Field Analogue of Brauer-Reynolds 69 13. Applications of GoN: Diophantine Approximation 71 13.1. Around Dirichlet's Theorem 71 13.2. The Best Possible One Variable Approximation Result 72 13.3. The Markoff Chain 73 14. Applications of GoN: Euclidean Rings 74 15. Applications of GoN: Representation Theorems for Quadratic Forms 76 15.1. Reminders on integral quadratic forms 76 15.2. An application of Hermite's Bound 80 15.3. The Two Squares Theorem 80 15.4. Binary Quadratic Forms 82 15.5. The Four Squares Theorem 87 2 2 2 2 15.6. The Quadratic Form x1 + ax2 + bx3 + abx4 89 15.7. Beyond Universal Forms 93 15.8. W´ojcik'sProof of the Three Squares Theorem 95 15.9. Ankeny's Proof of the Three Squares Theorem 99 GEOMETRY OF NUMBERS WITH APPLICATIONS TO NUMBER THEORY 3 15.10. Mordell's Proof of the Three Squares Theorem 101 15.11. Some applications of the Three Squares Theorem 103 15.12. The Ramanujan-Dickson Ternary Forms 104 16. Applications of GoN: Isotropic Vectors for Quadratic Forms 107 16.1. Cassels's Isotropy Theorem 107 16.2. Legendre's Theorem 108 16.3. Holzer's Theorem 117 16.4. The Cochrane-Mitchell Theorem 120 16.5. Nunley's Thesis 123 17. GoN Applied to Diophantine Equations Over Number Fields 123 17.1. Reminders on quadratic forms over number fields 123 17.2. Sums of Two Squares in Integral Domainsp 125 17.3. Sums of two squares in ZK , K = Q( 5) 128 17.4. Sums of Squares in Z[i] 130 17.5. Hermite constants in number fields 132 18. Geometry of Numbers Over Function Fields 133 18.1. No, seriously. 133 18.2. Tornheim's Linear Forms Theorem 133 18.3. Eichler's Linear Forms Theorem 136 18.4. Function Field Vinogradov Lemma 137 18.5. Prestel's Isotropy Theorem 138 18.6. The Prospect of a GoN Proof for Ternary Hasse-Minkowski 140 1 18.7. Chonoles's Geometry of Numbers in Fq(( t )) 142 18.8. Mahler's non-Archimedean Geometry of Numbers 145 18.9. Normed Rings 147 18.10. Gerstein-Quebbemann 149 19. Abstract Blichfeldt and Minkowski 152 References 154 1. Lattices in Euclidean Space Fix a positive integer N, and consider N-dimensional Euclidean space RN . In particular, under addition RN has the structure of a locally compact topological group. A vector group is a topological group G isomorphic to a subgroup of (RN ; +) for some N. In particular, RN is a vector group. 1.1. Discrete vector groups. At the other extreme are the discrete subgroups: those for which the induced subspace topology is the discrete topology. Exercise 1.1: Show that for a subgroup G ⊂ RN the following are equivalent: (i) The infimum of the lengths of all nonzero elements of G is positive. (ii) G is discrete. Proposition 1.1. Let G be a Hausdorff topological group and H a locally compact subgroup. Then H is closed in G. In particular, every discrete subgroup of a Hausdorff group is closed. 4 PETE L. CLARK Proof. Let K be a compact neighborhood of the identity in H. Let U be an open neighborhhod of the identity in G such that U \ H ⊂ K. Let x 2 H. Then there is a neighborhood V of x such that V −1V ⊂ U, so then (V \ H)−1(V \ H) ⊂ K: Since x 2 H, there exists y 2 V \ H, and then V \ H ⊂ yK. Since for every neighborhood W of x, W \ V is also a neighborhood of x and thus W \ V \ H 6= ;, x 2 V \ H. Since yK is compact in the Hausdorff space H, it is closed and thus x 2 V \ H ⊂ yK = yK ⊂ H. So H is closed. Example 1.2: For every 0 ≤ n ≤ N, there is a discrete subgroup of RN isomorphic as a group to Zn. This is almost obvious: the subgroup ZN of RN { what we call often call the standard integer lattice { is discrete, and hence so too are the subgroups Zn = ZN \ (Rn × 0N−n). Thus for all n ≤ N, there are discrete subgroups of RN which are, as abstract groups, free abelian of rank n. In fact the converse is also true: every discrete subgroup of RN is free abelian of rank at most N. But this is not obvious! The following exercise drives this home. 1 Exercise 1.3: a) Show that, as abstract groups, (RN ; +) =∼ (R; +). b) Deduce that for all n; N 2 Z+, RN admits a subgroup G =∼ Zn. c) Show in fact that RN admits a subgroup which is free abelian of rank κ for every cardinal number κ ≤ #R. Define the real rank r(G) of a vector group G ⊂ RN to be the maximal car- dinality of an R-linearly independent subset of G, so 0 ≤ r(G) ≤ N. For instance, the discrete subgroup Zn ⊂ ZN ⊂ RN of Example 1.1 above has real rank n. Theorem 1.2. Let G be a discrete subgroup of RN , of real rank r. There are R-linearly independent vectors v1; : : : ; vr such that G = hv1; : : : ; vriZ. Proof. By Proposition 1.1, G is closed.First observe that r = 0 () G = f0g and this is a trivial case. Henceforth we assume 0 < r ≤ N. By definition of real rank, there are e1; : : : ; er 2 G which are R-linearly independent. Let P = Pr f i=1 xiei j ei 2 [0; 1]g be the corresponding paralleletope. Thus G\P is a closed, discrete subspace of a compact set, hence finite. Let x 2 G. Since r is the real rank Pr of G, there are λ1; : : : ; λr 2 R such that x = i=1 λiei. For j 2 Z, put r X xj = jx − bjλicei: i=1 Thus r X xj = (jλi − bjλic)ei; i=1 Pr so xj 2 P \ G. Since x = x1 + i=1bλicei, we seee that G is generated as a Z-module by G \ P hence is finitely generated. Further, since G \ P is finite and Z is infinite, there are distinct j; k 2 Z such that xj = xk. Then (j − k)λi = bjλic − bkλic; 1We maintain our convention that N is an arbitrary, “fixed” positive integer.
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