# Women in Numbers 2: Research Directions in Number Theory

Share this page *Edited by *
*Chantal David; Matilde Lalín; Michelle Manes*

A co-publication of the AMS and Centre de Recherches Mathématiques

The second Women in Numbers workshop (WIN2) was held November 6–11,
2011, at the Banff International Research Station (BIRS) in Banff,
Alberta, Canada. During the workshop, group leaders presented open
problems in various areas of number theory, and working groups tackled
those problems in collaborations begun at the workshop and continuing
long after.

This volume collects articles written by participants of WIN2.
Survey papers written by project leaders are designed to introduce
areas of active research in number theory to advanced graduate
students and recent PhDs. Original research articles by the project
groups detail their work on the open problems tackled during and after
WIN2. Other articles in this volume contain new research on related
topics by women number theorists.

The articles collected here encompass a wide range of topics in
number theory including Galois representations, the Tamagawa number
conjecture, arithmetic intersection formulas, Mahler measures, Newton
polygons, the Dwork family, elliptic curves, cryptography, and
supercongruences.

WIN2 and this Proceedings volume are part of the Women in Numbers
network, aimed at increasing the visibility of women researchers'
contributions to number theory and at increasing the participation of
women mathematicians in number theory and related fields.

# Table of Contents

## Women in Numbers 2: Research Directions in Number Theory

- Preface vii8 free
- The Local Equivariant Tamagawa Number Conjecture for Almost Abelian Extensions 112 free
- Images of Metabelian Galois Representations Associated to Elliptic Curves 2940
- Newton Polygons for a Variant of the Kloosterman Family 4758
- Comparing Arithmetic Intersection Formulas for Denominators of Igusa Class Polynomials 6576
- An Algorithmic Approach to the Dwork Family 8394
- Ranks “Cheat Sheet” 101112
- 1. Mordell-Weil group, rank, and Tate-Shafarevich group 101112
- 2. 𝐿-function, analytic rank, and BSD (Birch and Swinnerton-Dyer) Conjecture 102113
- 3. (Un)boundedness 103114
- 4. Distribution 104115
- 5. Averages 104115
- 6. Parity 104115
- 7. Quadratic Twists 105116
- 8. Selmer Groups and Selmer Ranks 105116
- 9. Open Questions 107118
- Background Material 108119
- Surveys 108119
- References 108119

- Fully Homomorphic Encryption for Mathematicians 111122
- Mahler Measure of Multivariable Polynomials 125136
- 1. Definition of Mahler Measure and Lehmer’s question 125136
- 2. Mahler Measure in several variables 127138
- 3. Examples 128139
- 4. Other occurences of Mahler measure 129140
- 5. An algebraic integration for Mahler measure: the role of the regulator 132143
- 6. The measures of a family of genus-one curves 134145
- 7. Mahler measures of families of 𝐾3-surfaces 137148
- Acknowledgement 145156
- References 145156

- Mahler Measure of Some Singular 𝐾3-surfaces 149160
- 1. Introduction 149160
- 2. Background on 𝐾3-surfaces 152163
- 3. Main results and the general strategy for the proof 155166
- 4. The Transcendental Lattice and the Rank 155166
- 5. Relating the Mahler Measure to a newform 158169
- 6. Relating 𝐿(𝐓(𝐘),𝐬) to a newform 161172
- 7. Infinite section for 𝑌₁₈ 163174
- Acknowledgements 168179
- References 168179

- Distribution of Squarefree Values of Sequences Associated with Elliptic Curves 171182
- Recent Advances for Ramanujan Type Supercongruences 189200
- 1. Introduction 189200
- 2. Ramanujan type formulas for 1/𝜋 190201
- 3. Ramanujan supercongrences of van Hamme 195206
- 4. Proofs of van Hamme’s supercongruences 195206
- 5. Hypergeometric methods for van Hamme’s conjectures 198209
- 6. Ramanujan supercongruences arising from K3 surfaces 200211
- Acknowledgements 204215
- References 204215