Don't judge a stack of curves by its cover

• Andrea Di Lorenzo (Università di Pisa)

Room: Aula MN The stack of all Gorenstein, log-canonically polarized, marked curves of genus one is a non-separated stack, it does not have a good moduli space, and some people might therefore not like it. In this talk, I will explain why it's actually a really useful stack. Joint projects with Luca Battistella, and with Luca Battistella and Michele Pernice.

Logarithmic Barsotti-Tate groups and semistable representations

• Alessandra Bertapelle (Università degli Studi di Padova)

Room: Aula MN Let $k$ be a perfect field of positive characteristic $p>0$, $K_0$ the field of fractions of the ring of Witt vectors $W(k)$, $K$ a finite totally ramified extension of $K_0$, and $\mathcal O_K$ its ring of integers. It is known that the Tate module construction gives an equivalence between the category of Barsotti-Tate groups over $\mathcal O_K$ and the category ${\bf Rep}_{\mathbb Z_p}^{\rm crys,\{0,1\}}({\rm Gal}(\bar K/K))$ of crystalline Galois $\mathbb Z_p$-representations with Hodge-Tate weights in {0,1}. In this talk, we will explain how the previous equivalence can be extended to the categories of (dual representable) logarithmic Barsotti-Tate groups and semistable representations. This result is part of joint work with Shanwen Wang and Heer Zhao.

Torsion bounds for a fixed abelian variety and varying number field

• Davide Lombardo (Università di Pisa)

Room: Aula C05 Let $A$ be an abelian variety over a number field $K$. In this talk, I will discuss the question of bounding the size of the torsion subgroup of $A(L)$ as $L$ ranges over the finite extensions of $K$. One can ask for the optimal exponent $\beta_A$ such that for all $\varepsilon >0$ there is a constant $C_{A, \varepsilon}$ such that $$ \#A(L)_{\operatorname{tors}} \leq C_{A, \varepsilon} [L:K]^{\beta_A + \varepsilon} $$ for all finite extensions $L/K$. Hindry and Ratazzi have formulated a precise conjecture in this direction: the value of $\beta_A$ should be given by a simple formula involving the Mumford-Tate group of $A$. They also proved this conjecture for several classes of abelian varieties for which the Mumford-Tate conjecture is known to hold. In joint work with Samuel Le Fourn and David Zywina, we give an unconditional formula for the optimal exponent $\beta_A$ and show that it agrees with the conjectured one under the assumption of Mumford-Tate. Using this, we prove that the conjecture of Hindry and Ratazzi is equivalent to the Mumford-Tate conjecture for abelian varieties over number fields. We also give sharp lower bounds on the degrees of the extensions of $K$ generated by a single torsion point.

Existence of Quasi-Projective Moduli Spaces of Unstable Objects in Abelian Categories

• George Cooper (Scuola Normale Superiore, Pisa)

Room: Aula C05 For a fixed Rudakov-type stability condition on a $k$-linear, hom-finite abelian category $\mathcal{A}$, we explain how to construct coarse moduli spaces parametrising unstable objects $X \in \mathcal{A}$ of a fixed Harder--Narasimhan type, using a generalisation of Non-Reductive Geometric Invariant Theory (NRGIT) developed by Modin for actions of $[\mathbb{A}^1/\mathbb{G}_m]$ on algebraic stacks. The resulting moduli spaces are relatively quasi-projective over the product of the moduli spaces parametrising the Harder--Narasimhan subquotients. As special cases, we recover the results of Jackson, Hamilton, Qiao and Hoskins--Jackson, all involving NRGIT, concerning the existence of schematic coarse moduli spaces of Gieseker unstable coherent sheaves on a projective scheme and unstable Higgs bundles of Harder--Narasimhan length $2$ on a curve. This talk is based on joint ongoing work with Ludvig Modin.