Why should a general Diophantine equation have only finitely many solutions?

In this talk I will explain why a general polynomial in several variables with integer coefficients should have only finitely many integer zeroes.

Indeed, the Lang-Vojta conjecture predicts that, if a polynomial with integer coefficients defines a hyperbolic variety over the complex numbers, then the corresponding Diophantine equation should have only finitely many zeroes.

The interplay between the analytic and arithmetic aspects of a Diophantine equation are not fully understood yet, and the Lang-Vojta conjecture provides a guideline as to how part of the picture looks like.

Besides explaining the Lang-Vojta conjecture, I will also present a down-to-earth consequence of the Lang-Vojta conjecture: the set of equivalence classes of "smooth" homogeneous polynomials of fixed degree in a fixed number of variables is finite.

We will explain how to prove the latter finiteness statement in the case of homogeneous polynomials of degree six in four variables by associating functorially to each such polynomial a lattice of rank 52.