(joint work with Peter Scholze) In our joint work with Scholze we need to give a meaning to statements like "the stack of principal G-bundles on the curve is smooth of dimension 0" and construct "smooth perfectoid charts on it". The problem is that in the perfectoid world there is no infinitesimals and thus no Jacobian criterion that would allow us to define what is a smooth morphism. The good notion in this setting is the one of a cohomologically smooth morphism, a morphism that satisfies relative Poincaré duality. I will explain a Jacobian criterion of cohomological smoothness for moduli spaces of sections of smooth algebraic varieties over the curve that allows us to solve our problems.[-]

Finding an explicit isogeny between two given isogenous elliptic curves over a finite field is considered a hard problem, even for quantum computers. In 2011 this led Jao and De Feo to propose a key exchange protocol that became known as SIDH, shorthand for Supersingular Isogeny Diÿe-Hellman. The security of SIDH does not rely on a pure isogeny problem, due to certain 'auxiliary' elliptic curve points that are exchanged during the protocol (for constructive reasons). In this talk I will discuss a break of SIDH that was discovered in collaboration with Thomas Decru. The attack uses isogenies between abelian surfaces and exploits the aforementioned auxiliary points, so it does not break the pure isogeny problem. I will also discuss improvements of this attack due to Maino et al. and Robert, as well as a countermeasure by Fouotsa et al., along with breaks of this countermeasure in some special cases.[-]

It is classical that, for example, there is a simple abelian variety of dimension $4$ which is not the jacobian of any curve of genus $4$, and it is not hard to see that there is one defined over the field of all algebraic numbers $\overline{\bf Q}$. In $2012$ Chai and Oort asked if there is a simple abelian fourfold, defined over $\overline{\bf Q}$, which is not even isogenous to any jacobian. In the same year Tsimerman answered ''yes''. Recently Zannier and I have done this over the rationals $\bf Q$, and with ''yes, almost all''. In my talk I will explain ''almost all'' the concepts involved.[-]

Let X and Y be two curves over a common base field k. Then we can consider the Jacobians Jac (X) and Jac (Y). On the level of principally polarized abelian varieties, we can form the product Jac (X) x Jac (Y). A logical question is then whether there exists a curve Z over k such that Jac (Z) is (possibly up to twist) isogenous to Jac (X) x Jac (Y).Frey and Kani considered the case where X and Y both have genus 1. The current talk will consider the case where X and Y have genus 1 and 2, respectively, which was considered in joint work with Jeroen Hanselman and Sam Schiavone for the case of gluing along 2-torsion.We will give criteria for the curve Z to exist, and methods to find an equation if it does. The first of these uses interpolation, and also determines the relevant twisting scalar. It can be used to find a Jacobian over QQ that admits a rational 70-torsion point. The second method is more geometrically inspired and exploits the geometry of the Kummer surface of Y. Applications will be discussed in passing.[-]

The main result of the talk by X. Guitart in this conference classifies the 92 geometric endomorphism algebras that arise among geometrically split abelian surfaces defined over $\mathbb{Q}$. In this talk, we will explain how only 54 of them arise as geometric endomorphism algebras of Jacobians of genus 2 curves defined over $\mathbb{Q}$, and how the remaining 38 do not. In particular, we exhibit 38 abelian surfaces defined over $\mathbb{Q}$ that are not isogenous over an algebraic closure of $\mathbb{Q}$ to any Jacobian of a genus 2 curve defined over $\mathbb{Q}$.

This is a joint work with X. Guitart and E. Florit, that builds on examples supplied by N. Elkies and C. Ritzenthaler, and uses F. Narbonne's thesis in an essential way.[-]

The general principally polarized abelian variety of dimension at most five is known to be a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully concrete theory of algebraic curves. I will discuss recent breakthrough on finding a structure theorem for principally polarized abelian varieties of dimension six as Prym-Tyurin varieties associated to covers with $E_6$-monodromy, and the implications this uniformization result has on the geometry of the moduli space $A_6$. This is joint work with Alexeev, Donagi, Izadi and Ortega.[-]