Research directions in number theory. Women in numbers IV

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