# Luxembourg Number Theory Day 2022

The **Luxembourg Number Theory Day** will take place on December 16, 2022, on the Esch-Belval Campus of the University of Luxembourg.

A Webex link to attend the talks remotely will be sent upon request. Please let us also know if you plan to attend in person.

### Speakers

Lassina Dembelé (King's College London)

Harald Helfgott (CNRS, University of Göttingen)

Igor Shparlinski (University of New South Wales)

Lola Thompson (Utrecht University)

### Contacts

Antonella Perucca, Pietro Sgobba, Gabor Wiese. Email addresses: name.surname at uni.lu

### Location

All of the talks will take place in room **MNO 1.050** on the 1st floor of the Maison du Nombre at 6, Avenue de la Fonte, Esch-sur Alzette. You can see this on Google Maps or OpenStreetMap.

### Schedule

**Time** | **Speaker** | **Title** |

9:30-10:10 | Igor Shparlinski | *Integers of prescribed arithmethic structure in residue classes* |

10:10-10:30 | *Coffee break* | |

10:30-11:10 | Lassina Dembelé | *Hilbert-Siegel modular forms, congruences and applications to abelian surfaces* |

11:20-12:00 | Lola Thompson | |

12:00-13:00 | *Lunch break* | |

13:00-13:40 | Harald Helfgott | |

### Abstracts

**Igor Shparlinski** (University of New South Wales) *Integers of prescribed arithmethic structure in residue classes*
We give an overview of recent results about the distribution of some special integers
in residues classes modulo a large integer $q$. Questions of this type were introduced by
Erdos, Odlyzko and Sarkozy (1987), who considered products of two primes as a relaxation
of the classical question about the distribution of primes in residue classes. Since that time,
numerous variations have appeared for different sequences of integers. The types of
numbers we discuss include smooth, square-free, square-full and almost primes integers.

We also expose, without going into technical details, the wealth of different techniques behind these results: sieve methods, bounds of short Kloosterman sums, bounds of short character sums and many others.

**Lassina Dembelé** (King's College London) *Hilbert-Siegel modular forms, congruences and applications to abelian surfaces*

In this talk, we will explain how to compute congruences of Hilbert-Siegel modular forms. We will then discuss how to use these congruences to find abelian surfaces with everywhere good
reduction and trivial endomorphism rings.