**Time:** July 17-18, 2024

**Venue:** E4-233, Yungu Campus

**Wednesday July 17**

14:00—14:50 **Peyre Emmanuel**, Accumulating subsets and where to find them

15:00—15:30 Break

15:30—16:20 **Xu Fei**, Counting lattice points in central simple algebras with a given characteristic polynomial

16:30—17:20 **Wei Dasheng**, Fibration method with multiple fibers and strong Approximation

**Thursday July 18**

08:30—09:20 **Pazuki Fabien**, Drinfeld singular moduli and units

09:30—10:00 Break

10:00—10:50 **Amerik Ekaterina**, Potential density of rational points on the Hilbert cube of certain K3 surfaces

11:00—11:50 **Zeng Jiang**, Gamma-positivity of variations of Eulerian polynomials

**Amerik Ekaterina** (Higher School of Economics) Potential density of rational points on the Hilbert cube of certain K3 surfaces

This is a joint work in progress with my student Mikhail Lozhkin. Oguiso in 2009 has studied the following automorphism f of Hilb^2 of a K3 surface admitting two projective embeddings as a quartic: each of the two projective embeddings gives rise to an involution on Hilb^2 (the Beauville involution), and f is their product. At the time I made a remark that one could use f to prove the potential density of rational points on Hilb^2. In this talk, we will consider the product of two Beauville involutions on Hilb^3 of a surface admitting two projective embeddings of degree 6, and use it to prove the potential density.

**Pazuki Fabien** (University of Copenhagen) Drinfeld singular moduli and units

Masser asked whether a singular modulus, which is always an algebraic integer, can be a unit. Habegger (2015) first answered that there is at most finitely many singular moduli which are also units, which was generalised by Li (2021) to values of modular polynomials at singular moduli. Bilu, Habegger, Kühne (2020) finally proved that there is no singular modulus which is also a unit. This was generalised by Campagna (2021) to $S$-unit singular moduli, when $S$ is the infinite set of primes congruent to 1 modulo 3. In joint work with Anglès, Armana, Bosser, we show the following analogue for Drinfeld modules of rank 2: for any fixed prime power $q\geq2$, there is at most finitely many $\mathbb{F}_q[T]$-Drinfeld modules of rank 2 such that their $j$-invariant is both CM and a unit. We will present the result and discuss ideas of the proof, as well as connections with the question of effectivity in André-Oort.

**Peyre Emmanuel** (Université Grenoble Alpes) Accumulating subsets and where to find them

In Manin's program concerning the number of rational points of bounded height, a central rôle is played by the accumulating subsets: subsets with a negligible closure in the adelic space, but with a positive contribution to the number of rational points.

Locating such sets is crucial for the program and this talk will explain how an approach combining a kind of arithmetic class and freeness may help solving this problem.

**Wei Dasheng** (Chinese Academy of Science) Fibration method with multiple fibers and strong Approximation

We develop the fibration method for producing rational points on the total space with multiple fibers over the projective line. As its application, we prove strong approximation property of two classes of P^1-connceted varieties: the smooth locus of singular del Pezzo surfaces of degree at least 4 or the smooth locus of complete normal toric varieties. On the other hand, we study strong approximation property of the intersection of two affine quadrics. As its application, we get an unconditional result of fibration method of rank $4$ with multiple fibers or not. This is a joint work with Jie Xu and Yi Zhu.

**Xu Fei** (Capital Normal University) Counting lattice points in central simple algebras with a given characteristic polynomial

Eskin, Mozes and Shah determined an asymptotic formula for integral matrices with a given irreducible characteristic polynomial over Z. We’ll extend this result to a central simple algebra based on our previous work about counting integral points in homogeneous spaces. This is a joint work in progress with Jiaqi Xie.

**Zeng Jiang** (Université Lyon 1) Gamma-positivity of variations of Eulerian polynomials

Gamma-positivity is a property that polynomials with symmetric coefficients may have, which directly implies their unimodality. After a brief overview of the definitions and known facts, I will prove some refinements of this property for variants of Eulerian polynomials.