From Hilbert schemes of points on a smooth surface to cohomological Hall algebras

• Francesco Sala (Università di Pisa)

The present talk is devoted to a gentle introduction to the theory of cohomological Hall algebras (COHAs) and how they arose from the study of the topology of moduli spaces, such as Hilbert schemes of points on smooth complex surfaces. In the second part of the talk, I will address the problem of constructing new representations of COHAs by using torsion pairs.

On structure and slopes of Drinfeld cusp forms

• Andrea Bandini (Università di Pisa)

Drinfeld modular forms are the function field analogue of classical modular forms and have proven to be quite difficult to handle from the computational and structural viewpoint because of the positive characteristic setting. As an example we mention the absence of Petersson inner product, an issue which makes the definition of newforms still uncertain. We shall present a few conjectures and partial results on the structure of Drinfeld cusp forms mainly focusing on oldforms and newforms in low levels, slopes of eigenforms for the Hecke operators and, if time permits, on derivatives of quasi modular forms.

Fields of moduli and the arithmetic of quotient singularities

• Angelo Vistoli (SNS, Pisa)

Given a perfect field $k$ with algebraic closure $k'$ and a variety $X$ over $k'$, the field of moduli of $X$ is the subfield of $k'$ of elements fixed by elements $s$ of the Galois group of $k'$ over $k$ such that the twist $X_s$ is isomorphic to $X$. Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety $X$ with a smooth marked point $p$ that ensure that the pair $(X,p)$ is defined over its field of moduli.

Tate--Shafarevich groups over function fields of curves

• Tamás Szamuely (Università di Pisa)

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme G of finite type over the function field $K$ of $C$, one can consider the Tate--Shafarevich group given by isomorphism classes of $G$-torsors locally trivial at completions of $K$ associated with closed points of $C$. The classical case is when $k$ is finite and hence $K$ is a global field; in this case, the Tate--Shafarevich group is conjecturally finite. But more recently these groups have been studied over base fields $k$ of higher cohomological dimension, such as $p$-adic or number fields. We shall explain some of the known results about them, including a recent one obtained in collaboration with D. Harari (building on earlier work of Saïdi and Tamagawa): The Tate--Shafarevich group is finite when $G$ comes from a $k$-group scheme and $k$ is a number field.