From 0 to p and back

To be announced.

The idea of foliation on moduli spaces of abelian varieties in positive characteristic p, due to Frans Oort, was presented in the Texel '99 conference Moduli of abelian varieties. Leaves in a foliation come in two sorts: central leaves supported by prime-to-p Hecke symmetries, and isogeny leaves supported by p-adic Hecke correspondences.

We will survey subsequent developments related to the notion of central leaves. These include: (a) the notion of sustained p-divisible groups, which is an updated version of the notion of geometrically fiberwise constant p-divisible groups, (b) the notion of Tate-linear formal varieties, with formal completions of central leaves as prominent examples, and (c) orbital rigidity of Tate-linear formal varieties, extending orbital rigidity of p-divisible groups.

The conjecture of Oort that each prime-to-p Hecke orbit is Zariski dense in the underlying central leaf has been confirmed, with the help of the above tools. We will discuss some further challenges, in the form of questions and conjectures.

The Brauer group of an algebraic variety is a group with many applications, in particular to the study of rational points. For a K3 surface over a number field, the transcendental part of its Brauer group is finite. It was shown by Cadoret-Charles that the size of its p primary torsion is uniformly bounded for K3 surfaces in one-dimensional families. We give a different proof of this result for one-dimensional families of K3 surfaces with a polarization by a fixed lattice. To be precise, we construct moduli spaces of K3 surfaces with a lattice polarization and a Brauer class, and use the geometry of their complex points to prove boundedness of Brauer groups for the K3 surfaces they parametrize. I will explain the construction and give a sketch of the proof of our boundedness result. This is joint work with D. Bragg and A. Várilly-Alvarado.

Honda-Tate theory describes abelian varieties over finite fields up to isogeny by Weil numbers. We will report about ongoing work aiming to enhance various recent descriptions of the category of abelian varieties over finite fields (without inverting isogenies) in terms of (non-)commutative algebra.
This is joint work with Tommaso Centeleghe.

We disprove the integral Hodge conjecture for abelian varieties and show that very general cubic threefolds are not stably rational. More precisely, we prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least four over the complex numbers is an even multiple of the minimal class, and that the same holds for the intermediate Jacobian of a very general cubic threefold. This is joint work with Philip Engel and Stefan Schreieder.

In this talk, we will discuss which endomorphisms and automorphisms abelian varieties can have, with a special focus on supersingular abelian varieties. More precisely, in the first part, we will present recent results from joint work with Tamagawa and Yu on realisability of orders as endomorphism rings of abelian varieties over algebraically closed fields, that build on previous work of Oort, Oort-van der Put and Oort-Zarhin on realisability of algebras as endomorphism algebras. For the second part, let S_g be the supersingular locus of the moduli space A_g over F_p of g-dimensional principally polarised abelian varieties, where p is a prime and g ≥ 1. We will use geometric stratifications on A_g and S_g, including the Ekedahl-Oort stratification, to explain the current status of Oort’s conjecture, which states that all generic points of S_g have automorphism group {±1}. This part is based on joint work with Yu.