Seminario Aritmética y Geometría en Valparaíso

Octubre

Martes 29 de Octubre, 15:00-17:00

Expositor: Gabriele Ranieri (PUCV)

Título: On the local-global divisibility principle over elliptic curves and other algebraic groups

Resumen:

Let $k$ be a  number field and let $A$ be a commutative algebraic group defined over $latex k$. Consider the following question:

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

This problem is called Local-global divisibility problem by $latex q$ on $A$ over $latex k$.
If for every $latex P \in A ( k )$ the answer to the Local-global divisibility problem is positive, we say that the Local-global divisibility principle for divisibility by $latex q$ holds for $A$ over $latex k$.

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 proved that there exists a constant $latex C ( [k: \mathbb{Q}] )$ depending just on $latex [k: \mathbb{Q}]$ such that, for every prime number $latex p > C ( [k: \mathbb{Q}] )$, for every elliptic curve $E$ defined over $latex k$, the Local-global divisibility principle for divisibility by $latex p^n$ holds for $E$ over $latex k$, for every positive integer $latex n$.

We give an idea of the proof of our result and we explain some possible relations between the Local-global divisibility problem and the non-triviality of the Tate-Shafarevich group.

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