Research Directions in Number Theory: Women in Numbers IV (Association for Women in Mathematics Series Book 19)

"These proceedings collect several number theory articles, most of which were written in connection to the workshop WIN4: Women in Numbers, held in August 2017, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. It collects papers disseminating research outcomes from collaborations initiated during the workshop as well as other original research contributions involving participants of the WIN workshops. The workshop and this volume are part of the WIN network, aimed at highlighting the research of women and gender minorities in number theory as well as increasing their participation and boosting their potential collaborations in number theory and related fields." - Prové de l'editor Preface......Page 7 Participants and Affiliations at the Time of the Workshop......Page 9 Workshop Website......Page 11 Acknowledgments......Page 12 Contents......Page 13 List of Contributors......Page 14 Ramanujan Graphs in Cryptography......Page 17 1 Introduction......Page 18 Part 1: Cryptographic Applications of Supersingular Isogeny Graphs......Page 20 2 Preliminaries......Page 21 3 The defeo:2014 Key-Exchange......Page 24 3.1 Hardness Assumptions......Page 25 4 Composing Isogenies......Page 28 Part 2: Constructions of Ramanujan Graphs......Page 32 5 Background on Ramanujan Graphs and Adèles......Page 34 6 LPS Graphs......Page 35 6.2 Infinite Tree of Lattices......Page 36 6.3 Hamiltonian Quaternions over a Local Field......Page 38 6.4 Explicit Isomorphism Between Generating Sets......Page 40 7 Strong Approximation......Page 43 7.1 Approximation for Invertible Quaternions......Page 44 7.2 Strong Approximation for LPS Graphs......Page 45 7.3 Strong Approximation for Pizer Graphs......Page 47 8 Pizer Graphs......Page 48 8.1 Overview of the Construction......Page 49 8.2 The Size of a Six-Regular Pizer Graph......Page 51 9 Relationship Between LPS and Pizer Constructions......Page 53 References......Page 55 1 Introduction......Page 57 2.1 Definitions and Properties......Page 59 3.1 Quaternion Algebras......Page 63 3.2 Computing Endomorphism Rings from Cycles in the -Isogeny Graphs......Page 64 4 A Condition for Linear Independence......Page 66 5 An Obstruction to Generating the Full Endomorphism Ring......Page 71 6 Examples......Page 72 Appendix A: Modified Schoof's Algorithm for Traces of Arbitrary Endomorphisms......Page 77 A.1. Complexity of Computing Endomorphisms on m-Torsion......Page 78 A.2. Computing the Trace on m-Torsion Points......Page 79 A.2.2. Complexity Analysis for Computing the Trace......Page 80 References......Page 81 Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves......Page 83 1 Introduction......Page 84 2.1 Chabauty–Coleman Method......Page 86 2.2.1 Coleman Integral Within a Residue Disk......Page 88 2.2.2 Coleman Integral Between Different Residue Disks......Page 89 3.1 Upper Bounds in Residue Disks......Page 90 3.2 Roots of p-Adic Power Series......Page 93 3.3 Outline of the Algorithm......Page 95 3.4.1 What If We Do Not Know P0C(Q) Such That [P0-∞]J(Q)`3́9`42`"̇613A``45`47`"603Ators?......Page 98 4 Curve Analysis......Page 99 4.1 Examples......Page 100 4.2 Variation of the Prime and the Strong Chabauty Conjecture......Page 104 References......Page 105 1 Introduction......Page 107 2 General Results on Elliptic K3 Surfaces......Page 108 4 The Modular Fibration and the Graph Gr......Page 110 5.1 Graph Method......Page 115 6.1 Fibration #1......Page 116 6.2 Fibration #51......Page 119 6.3 Fibration #8......Page 120 7 From Fibration #8 to Fibrations with Singular Fibers of Type I4 and Others......Page 121 7.1 From Fibration #8 to Fibration #33......Page 122 8 From Fibration #33......Page 124 9 From Fibration #26......Page 125 10 Fibration of Rank 0......Page 126 11 Pairs of Fibrations with the Same Trivial Lattice......Page 127 References......Page 129 Newton Polygons of Cyclic Covers of the Projective Line Branched at Three Points......Page 131 1 Introduction......Page 132 Organization of the Paper......Page 133 2.1 The Group Algebra Q[μm]......Page 134 2.2 Cyclic Covers of the Projective Line......Page 135 2.3 Newton Polygons......Page 136 3 Newton Polygons of Curves with Complex Multiplication......Page 137 3.1 Shimura–Taniyama Method......Page 138 3.2 Slopes with Large Denominators......Page 140 4 Tables......Page 142 5 Applications......Page 144 References......Page 147 Arboreal Representations for Rational Maps with Few CriticalPoints......Page 149 1 Introduction......Page 150 2 Toward a Serre-Type Open Image Theorem for Arboreal Representations......Page 153 3.1 Cubic Polynomial Examples......Page 161 3.2 A Family of Degree 2 Rational Maps......Page 162 References......Page 166 1 Introduction......Page 168 2 Dessins for Single-Cycle Belyi Maps......Page 171 3.1 Single-Cycle Polynomials......Page 172 3.2 Symmetric Single-Cycle Belyi Maps......Page 173 References......Page 174 1 Introduction......Page 175 2 Preliminaries......Page 177 3 The Density as an Infinite Sum......Page 178 4 General Remarks......Page 179 5 Examples and Special Cases......Page 181 6 On the Computability of the Density......Page 183 References......Page 185 Quantum Modular Forms and Singular Combinatorial Serieswith Distinct Roots of Unity......Page 186 1.1 Background......Page 187 1.2 Quantum Modular Forms......Page 192 1.3 Results......Page 193 2.1 Modular, Mock Modular, and Jacobi Forms......Page 195 3 The Quantum Set......Page 199 4 Proof of Theorem 1.7......Page 202 References......Page 207