# Seminario Aritmética y Geometría en Valparaíso

## Noviembre

Martes  18 de Noviembre,  sala 2-3, IMA de la PUCV, 15:40-16:40

Expositor: Pierre Gillibert (PUCV)

Título:  Kuratowski’s characterisation of the Aleph

Resumen: I shall investigate an infinite combinatorial statement given by Kuratowski to characterize (small) infinite cardinals. After some historical background that give insight, and yeld to this statement, I shall show relations with some undecidable statements and talk about generalisations.

Martes  4 de Noviembre,  sala 2-3, IMA de la PUCV, 15:40-17:00

Expositor: Gabriele Ranieri (PUCV)

Resumen:   Let $k$  be a number field and let $\mathcal{A}$ be a
commutative algebraic group defined over $k$. Consider the following
question:

Problem. Let $P \in \mathcal{A} ( k )$ and let $q$ be a positive
integer. Suppose that for all but finitely many places $v$ of $k$,
there exists $D_v \in \mathcal{A} ( k_v )$ such that $P = q D_v$. Does
there exist $D \in \mathcal{A} ( k )$ such that $P = qD$?

This problem is called Local-global divisibility problem by $q$ on
$\mathcal{A} ( k )$.

Dvornicich and Zannier gave a cohomological interpretation of the
Local-global divisibility problem.
By using this interpretation, in two joint works with Laura Paladino
(University of Calabria) and Evelina Viada (University of Goettingen),
we studied the problem in the case when $\mathcal{A}$ is an elliptic
curve.
Recently, we have partially extended our results on elliptic curves to
the more general family of $GL_2$-type varieties.

We give an idea of the proof of our results and we explain some other
possible generalizations.