**Contemporary Mathematics**

Volume: 703;
2018;
222 pp;
Softcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-2856-3
Product Code: CONM/703**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-4674-1
Product Code: CONM/703.E**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

# Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

Share this page *Edited by *
*Andreas Malmendier; Tony Shaska*

This volume contains the proceedings of the AMS Special Session on
Higher Genus Curves and Fibrations in Mathematical Physics and
Arithmetic Geometry, held on January 8, 2016, in Seattle,
Washington.

Algebraic curves and their fibrations have played a major role in
both mathematical physics and arithmetic geometry. This volume focuses
on the role of higher genus curves; in particular, hyperelliptic and
superelliptic curves in algebraic geometry and mathematical
physics.

The articles in this volume investigate the automorphism groups of
curves and superelliptic curves and results regarding integral points
on curves and their applications in mirror symmetry. Moreover,
geometric subjects are addressed, such as elliptic \(K\)3 surfaces over
the rationals, the birational type of Hurwitz spaces, and links
between projective geometry and abelian functions.

#### Readership

Graduate students and research mathematicians interested in superelliptic and hypelliptic curves.

# Table of Contents

## Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- A lower bound for the number of finitely maximal 𝐶_{𝑝}-actions on a compact oriented surface 110
- Galois action on regular dessins d’enfant with simple group action 1322
- Equations of Riemann surfaces with automorphisms 3342
- On the field of moduli of superelliptic curves 4756
- Minimal integral Weierstrass equations for genus 2 curves 6372
- 1. Introduction 6372
- 2. Reduction of binary quintics and sextics 6574
- 3. Julia quadratic of genus two curves with extra automorphisms 7281
- 4. Minimal models of curves with extra involutions 7584
- 5. Some heuristics for curves with extra involutions defined over Q 7887
- Acknowledgments 8291
- References 8291

- Rational points in the moduli space of genus two 8392
- 1. Introduction 8392
- 2. A database of integral binary sextics 8594
- 3. Heights of genus two curves 8695
- 4. Genus 2 curves over C 8897
- 5. Algebraic invariants 91100
- 6. Automorphisms 97106
- 7. Genus 2 curves defined over Q 100109
- 8. Minimal discriminant for Weierstrass equations 103112
- 9. Constructing the databases 106115
- Appendix A. Functions of the genus 2 package 110119
- Genus 2 package 110119
- Moduli Space 110119
- Creating the databases 110119
- Appendix B. Basic Invariants and relations among them 111120
- References 113122

- Strong arithmetic mirror symmetry and toric isogenies 117126
- Inose’s construction and elliptic 𝐾3 surfaces with Mordell-Weil rank 15 revisited 131140
- Higher-order Weierstrass weights of branch points on superelliptic curves 143152
- Poncelet’s porism and projective fibrations 157166
- Extending Runge’s method for integral points 171180
- Self-inversive polynomials, curves, and codes 189198
- Syzygy divisors on Hurwitz spaces 209218
- Back Cover Back Cover1234