Uniformity Results in Diophantine Geometry

Amos Turchet

http://www.math.chalmers.se/~tamos/

Date(s) : 25/01/2016   iCal
14 h 00 min - 15 h 00 min

In 1997 Caporaso, Harris and Mazur proved that Lang Conjecture (i.e. rational points in general type varieties are not Zariski dense) implies that the number of rational point in curves of genus > 1 are not only finite (Falting’s Theorem) but uniform; in particular there exists a bound for their number depending only on the genus and on the base field. This result has been extended to surfaces of general type by work of Hassett. Analogous problems have been treated for (stably) integral points – introduced by Abramovich – for elliptic curves and principally polarised abelian varieties, where uniformity, conditionally on Lang-Volta Conjecture, has been proved to hold by work of Abramovich and Abramovich-Matzuki. The focus of the talk will be an introduction to the subjects and its links with questions of positivity of bundles and the theory of stable pairs. If time permits I will report on a work-in-progress project, joint with Kenneth Ascher, aiming to extend the results for integral points to all log general type surfaces.

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