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Lang-Vojta conjecture and smooth hypersurfaces over number fields

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Ariyan Javanpeykar (Mainz)

What
  • Seminar
When May 12, 2016
from 10:30 am to 11:30 am
Where Bonn, Hörsaal MPI, Vivatsgasse 7
Contact Name sachinidis
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Siegel proved the finiteness of the number of solutions to the unit equation in a number ring, i.e., for a number field K with ring of integers O, the equation x+y = 1 has only finitely many solutions in O*. That is, reformulated in more algebro-geometric terms, the hyperbolic curve P^1-{0,1,infty} has only finitely many "integral points". In 1983, Faltings proved the Mordell conjecture generalizing Siegel's theorem: a hyperbolic complex algebraic curve has only finitely many integral points. Inspired by Faltings's and Siegel's finiteness results, Lang and Vojta formulated a general finiteness conjecture for "integral points" on complex algebraic varieties: a hyperbolic complex algebraic variety has only finitely many "integral points" . In this talk we will explain the Lang-Vojta conjecture and we will explain some of its consequences for the arithmetic of homogeneous polynomials over number fields. This is joint work with Daniel Loughran.