Women in Numbers Europe II: Contributions to Number Theory and Arithmetic Geometry (Association for Women in Mathematics Series, 11)

Annotation Inspired by the September 2016 conference of the same name, this second volume highlights recent research in a wide range of topics in contemporary number theory and arithmetic geometry. Research reports from projects started at the conference, expository papers describing ongoing research, and contributed papers from women number theorists outside the conference make up this diverse volume. Topics cover a broad range of topics such as arithmetic dynamics, failure of local-global principles, geometry in positive characteristics, and heights of algebraic integers. The use of tools from algebra, analysis and geometry, as well as computational methods exemplifies the wealth of techniques available to modern researchers in number theory. Exploring connections between different branches of mathematics and combining different points of view, these papers continue the tradition of supporting and highlighting the contributions of women number theorists at a variety of career stages. Perfect for students and researchers interested in the field, this volume provides an easily accessible introduction and has the potential to inspire future work Preface 7 Acknowledgement 8 Contents 9 Lower Bounds for Heights in Relative Galois Extensions 11 1 Introduction 12 2 Preliminaries 14 2.1 Finite Linear Groups 14 2.2 Height of Algebraic Numbers 15 2.3 Estimates for φ(n)/n 16 3 Some Useful Lemmas 18 4 Proof of Theorem 1 21 5 Proof of Theorem 2 24 References 27 Reductions of Algebraic Integers II 28 1 Introduction 28 2 Preliminaries on the Chebotarev Density Theorem 30 3 Cyclotomic and Kummer Extensions 32 4 Prescribed Torsion in the Reductions 34 5 General Formulas for the Density 35 6 Formulas to Reduce to Known Cases 38 7 Examples 40 References 42 Reductions of One-Dimensional Tori II 43 References 45 On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments 46 1 Introduction 47 2 The Carlitz Rank: Basic Properties 48 3 On the Difference of Permutation Polynomials 52 4 On Iterations of Permutation Polynomials 55 References 61 Dynamical Belyi Maps 63 1 Introduction 64 1.1 Outline and Summary of Results 64 2 Background on Belyi Maps 65 3 Families of Dynamical Belyi Maps 69 4 Reduction Properties of Normalized Belyi Maps 74 5 Good Inseparable Monomial Reduction 78 6 Dynamics 83 References 87 Discriminant Twins 89 1 Statement of Theorem 89 2 Elliptic Curves of Prime Conductor 90 3 p-Adic Uniformization of Elliptic Curves and Isogenies 92 4 Infinitely Many Discriminant Twin Pairs 94 5 Semistable Isogenous Curves 95 6 Main Theorem 97 7 Number Fields 102 8 Semistable Non-isogenous Discriminant Twins 103 Appendix 105 5-Isogenies 105 3-Isogenies 107 2-Isogenies 109 References 112 The a-Number of Hyperelliptic Curves 113 1 Introduction 113 2 Background Information 115 2.1 The Cartier Operator 115 2.2 p-Rank and a-Number 116 3 Results 117 4 Computations and Examples for Small Primes 119 4.1 For p=3 119 4.2 For p=5 119 4.3 For p=7 120 5 Further Lowering the Bound 120 6 Sage Code 121 References 122 Non-ordinary Curves with a Prym Variety of Low p-Rank 123 1 Introduction 124 1.1 Outline of the Paper 127 2 Background 127 2.1 Notation 127 2.2 The Cartier-Manin Matrix 127 2.2.1 The Cartier-Manin Matrix of a Hyperelliptic Curve 128 2.3 The Hasse-Witt Matrix 128 2.4 The p-Rank 129 2.5 Prym Varieties 130 2.6 Moduli Spaces 130 3 Hasse-Witt Matrices of Genus 3 Curves and Their Prym Varieties 131 3.1 The Prym Variety of an Unramified Double Cover of a Plane Quartic 131 3.2 Hasse-Witt Matrices 132 3.3 The p-Ranks of X and Z 133 4 The Hasse-Witt Matrix of a Smooth Plane Quartic Defined as an Intersection in P3 138 5 The Fiber of the Prym Map When g=3 141 5.1 Review of Work of Verra 142 5.2 Explicit Version of the Fiber of the Prym Map 143 5.2.1 The Kummer Surface 143 5.2.2 Plane Quartics as Planar Sections of the Kummer Surface 145 5.3 The Hasse-Witt Matrix of X 145 5.4 An Existence Result for Each p 5 -5mumod5mu-6 146 5.5 The Condition that X Is Non-ordinary 150 5.5.1 The Condition that X Is Non-ordinary 150 5.5.2 The Condition that X Is Non-ordinary and Z Has p-Rank 1 150 5.5.3 The Condition that X Is Non-ordinary and Z Has p-Rank 0 151 5.5.4 Example: When p=3 152 5.5.5 Example: Genus 3 Curves Having Pryms of 3-Rank 1 When p=3 152 5.6 The Moduli Space of Genus 3 Curves Having Pryms of 3-Rank 0 When p=3 153 5.6.1 A Family of Genus 2 Curves with 3-Rank 0 When p=3 153 5.6.2 The Locus of the Parameter Space Where X Is Non-ordinary When p=3 154 6 Points on the Kummer Surface 156 7 Results for Arbitrary g 158 7.1 Increasing the p-Rank of the Prym Variety 158 7.2 Background on Boundary of Rg 158 7.3 Some Extra Results When p=3 160 7.4 A Dimension Result 161 7.5 Final Result 162 References 164 Elliptic Fibrations on Covers of the Elliptic Modular Surface of Level 5 165 1 Introduction 166 1.1 K3 Surfaces Arising from Rational Elliptic Surfaces and Their Elliptic Fibrations 166 1.2 Outline of the Paper 167 2 The Surface R5,5 168 3 Conic Bundles on R5,5 171 4 K3 Surfaces Obtained by R5,5 174 4.1 The Branch Fibers Are 2I5: The K3 Surface S5,5 174 4.1.1 The Surface R"0365R 174 4.1.2 Geometric Description of S5,5 and Its Néron–Severi Group 175 4.1.3 Weierstrass Equation of S5,5 177 4.1.4 Double Cover of P2 177 4.2 The Branch Fibers Are 2I0: The K3 Surface X5,5 178 4.2.1 Geometric Description of X5,5 and Its Néron–Severi Group 178 4.2.2 Weierstrass Equation of X5,5 178 4.2.3 Double Cover of P2 179 4.3 Branch Fibers I5 and I1 179 4.4 Branch Fibers I5 and I0 180 4.5 Branch Fibers I1 and I0 181 4.6 Branch Fibers 2I1 181 5 Elliptic Fibrations on K3 Surfaces Induced by the Conic Bundles 181 5.1 An Example 182 5.1.1 Equation of the Elliptic Fibration on S5,5 Induced by |B1| 182 5.2 An Algorithm to Compute Weierstrass Equations 183 5.3 The Elliptic Fibrations Induced by Conic Bundles 185 5.3.1 The K3 Surface S5,5 185 5.3.2 The K3 Surface X5,5 186 5.3.3 Other K3 Surfaces 186 6 The K3 Surface S5,5 and Its Elliptic Fibrations 187 6.1 Splitting Genus 1 Fibrations 189 6.1.1 An Example, the Fibration 6 189 6.1.2 Splitting Genus 1 Fibration: An Algorithm 191 6.2 The Elliptic Fibrations on S5,5 192 6.2.1 Elliptic Fibrations Induced by Splitting Genus 1 Pencils 193 6.2.2 Elliptic Fibrations Induced by Generalized Conic Bundles 194 7 The K3 Surface X5,5 and Its Elliptic Fibrations 197 7.1 The List of All the Elliptic Fibrations 198 7.2 Fibration of Type 3: An Example, the Fibration 26 200 References 202 On Birch and Swinnerton-Dyer's Cubic Surfaces 204 1 Introduction 204 1.1 The Hasse Principle for Cubic Surfaces 205 1.2 Outline 206 2 Setup and Background 206 3 Proof of Main Theorem 207 4 Invariant Map Computations 211 5 Examples 214 Appendix 215 References 217