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Variétés rationnelles

Horaires : Le vendredi 20 mai 2016, 15h15-16h15

Lieu : ENS, salle W

The Lang-Vojta conjecture and smooth hypersurfaces over number fields.

Ariyan Javanpeykar (Mayence)

Siegel proved the finiteness of the set 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,infinite} 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 start by explaining the Lang-Vojta conjecture and then proceed to prove some of its consequences for the arithmetic of homogeneous polynomials over number fields. This is joint work with Daniel Loughran.


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