seminar: On the Hasse principle for the division of points in a commutative algebraic group

speaker: Roberto Dvornicich (Università di Pisa)

abstract: Let $\A$ be a commutative algebraic group defined over a number field $k$. We consider the following question: {\sl Let $r$ be a positive integer and let $P\in \A(k)$. Suppose that for all but a finite number of primes $v$ of $k$ we have $P=rD_v$, for some $D_v\in \A(k_v)$. Can one conclude that there exists $D\in \A(k)$ such that $P=rD$?} A complete answer for the case of the multiplicative group $\G_m$ is classical. We study other instances, mainly concerning elliptic curves and algebraic tori, obtaining results in both directions: namely, we have families of examples for which the answer is positive and families of examples for which the answer is negative. This is a joint work with U. Zannier.

timetable:
Wed 25 May, 11:00 - 12:00, Sala Conferenze Centro De Giorgi
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