A CIMPA school on

# Lattices, Heights and Diophantine Approximation

Urgench State University, Urgench, Uzbekistan

June 24th - July 5th 2024

## About Us

The school aims at introducing the students to a few lines of research in number theory revolving around the concepts of lattices, heights and diophantine approximation. Part of the school is devoted to presenting the recent proof by Maryna Viazovska that the densest sphere packing in dimension 8 is the one given by the E_8 lattice. In 2022 she was awarded the Fields medal. To this end we introduce the students to lattices, sphere packings and modular forms. We will prove the Cohn-Elkies bound and finally we will study Viazovska's construction of the function which optimizes such bound and proves that the E_8 lattice is the densest packing in dimension 8. In the other part of the program we will introduce the machinery of heights, which are standard tools of Diophantine geometry used to measure arithmetic complexity of objects. We will then demonstrate the use of height functions in Diophantine approximation and Diophantine geometry discussing results and conjectures such as the Mordell-Weil theorem, Faltings' theorem, Siegel's lemma, Cassels' theorem, Lehmer's conjecture, and many others. On the Diophantine approximation side, we will discuss the central themes of equidistribution and approximation of reals by algebraic numbers. We expect the participants to gain familiarity with these central and vibrant areas of mathematics that have been at the forefront of mathematical research for over a hundred years. Our program is specifically designed in a way that assumes a rather modest background, but with a concentrated and focused approach takes the audience to some of the modern-day research directions.

## Team

#### Elisa Lorenzo Garcia

External coordinatorUniversité de Neuchâtel & Université de Rennes 1

#### Zafar Ibragimov

Local coordinatorUrgench State University

Scientific Committee

- Lenny Fukshansky (Claremont McKenna College)
- Elisa Lorenzo Garcia (Université de Neuchâtel & Université de Rennes 1)
- Kate Petersen (University of Minnesota Duluth)
- Valerio Talamanca (Università Roma Tre)

Organizing Committee

- Bakhrom Abdullaev (Urgench State University)
- Alimardon Atamuratov (V.I.Romanovsky Institute of mathematics)
- Aygul Babajanova (V.I.Romanovsky Institute of mathematics)
- Nargiza Boboyorova (Urgench State University)
- Zafar Ibragimov, (Urgench State University)
- Jumanazar Khujamov (Urgench State University)
- Gayrat Urazboev (Urgench State University)
- Mokhira Vaisova (Urgench State University)

### Courses

### 01 Heights of algebraic numbers

**Michel Waldschmidt**

Videos of introductory classes to be watched by participants before the school:

Class 1,
Class 2,
Class 3,
Class 4,
References

### 02 Diophantine geometry

Elisa Lorenzo Garcia

### 03 Diophantine approximation

Oleg German

### 04 Equidistribution

Kate Petersen

### 05 Search bounds for Diophantine equations

Lenny Fukshansky

### 06 Introduction to sphere packing

Peter Stevenhagen

### 07 Introduction to modular forms

Francesco Pappalardi

### 08 Cohn-Elkies bounds

René Schoof

### 09 The densest packing in dimension 8

Laura Geatti

**References:**

H. Cohn, N. Elkies, * New upper bounds on sphere packings I, * Ann. Math., ** 157** (2003), 689-714.

H. Cohn, *A conceptual breakthrough in sphere packing,* Notices A.M.S., Providence 2017. pdf

J. Oesterlé, * Densité maximale des empilements de sphères en dimension 8 et 24, * Sém. Bourbaki, 2016-17, n. 1133. pdf

A. Slipper, * Modular magic,* Bach. of Arts thesis, Harvard 2018. pdf

M. S. Viazovska, * The sphere packing problem in dimension 8, * Ann. Math., **185 ** (2017), 991-1015. pdf

D. Zagier, * Elliptic Modular Forms and Their Applications, * In The 1-2-3 of Modular Forms, Universitext, Springer 2008 pdf

**Useful links:**

J. Oesterlé talk at Seminaire Bourbaki 17/06/2017 video

## Lecturers

#### Lenny Fukshansky

Claremont McKenna CollegeUnited States of America

#### Elisa Lorenzo Garcia

Université de NeuchâtelSwitzerland

#### Laura Geatti

Università di Roma Tor VergataItaly

#### Oleg German

Lomonosov Moscow State UniversityRussia

#### Francesco Pappalardi

Università Roma TreItaly

#### Kate Petersen

University of Minnesota DuluthUnited States of America

#### René Schoof

Università di Roma Tor VergataItaly

#### Peter Stevenhagen

Universiteit LeidenThe Netherlands

#### Michel Waldschmidt

Sorbonne UniversitéFrance

## Schedule

### First week

Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|

08:00-8:50 | Opening Ceremony | ||||

09:00-9:50 | Sphere packing | Diophantine geometry | Heights of numbers | Modular forms | Cohn-Elkies bounds |

10:00-10:50 | Heights of numbers | Modular forms | Cohn-Elkies bounds | Heights of numbers | Sphere packing |

10:50-11:10 | Coffee break | ||||

11:10-12:00 | Modular forms | Cohn-Elkies bounds | Sphere packing | Diophantine geometry | Heights of numbers |

12:10-13:00 | Diophantine geometry | Sphere packing | Modular forms | Cohn-Elkies bounds | Diophantine geometry |

13:00-16:30 | Lunch break | ||||

16:30-17:20 | Cohn-Elkies bounds | Heights of numbers | Free Afternoon | Sphere packing | Modular forms |

17:30-18:20 | Sphere packing | Modular forms | Diophantine geometry | Cohn-Elkies bounds |

### Second week

Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|

09:00-9:50 | Equidistribution | Search bounds | Densest packing | Diophantine approximation | Equidistribution |

10:00-10:50 | Search bounds | Diophantine approximation | Equidistribution | Densest packing | Search bounds |

10:50-11:10 | Coffee break | ||||

11:10-12:00 | Diophantine approximation | Densest packing | Search bounds | Equidistribution | Diophantine approximation |

12:10-13:00 | Densest packing | Equidistribution | Diophantine approximation | Search bounds | Densest packing |

13:00-16:30 | Lunch break | ||||

16:30-17:20 | Diophantine geometry | Search bounds | Free Afternoon | Equidistribution | |

17:30-18:20 | Heights of numbers | Diophantine approximation | Densest packing |

## Photogallery

- All
- A CIMPA School - 2021
- A WAMS School - 2022
- A CIMPA School - 2024

## Contact

### Address

14 Khamid Alimdjan Street, Urgench, Uzbekistan