Rational points on Châtelet surfaces

This research talk is about an upcoming joint paper with Alexei Skorobogatov.

Châtelet surfaces of degree $d$ are surfaces of the form $x^2 -a y^2 =f(t)$, where $f$ is a fixed integer polynomial of degree $d$ and $a$ is a fixed non-square integer. When $f$ has degree up to 4 (or when $f$ is a product of integer linear polynomials ) it has been shown that the Brauer-Manin obstruction is the only one to the Hasse principle. This is the result of decades of investigations by Swinnerton-Dyer, Colliot-Thélène, Skorobogatov, Browning and Matthiesen, among others.

Going beyond degree 4 for polynomials of general type has been a very popular question which has seen no progress in the last decades. We use techniques from analytic number theory, related to equidistribution of the Möbius function, to prove that for 100% of all polynomials $f$ (ordered by coefficients) gives Châtelet surfaces with a rational point. In particular, for any degree $d>4$, 100% of these surfaces satisfy the Hasse principle.