Ariyan Javanpeykar: The Lang–Vojta conjecture and the moduli of smooth hypersurfaces

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,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 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.