Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Vladimir Mitankin
Affiliation:

Universidade Federal do Rio de Janeiro
Date:

Wed, 2018-01-10 14:30 - 15:30 Building up on the work of Colliot-Thélène and Sansuc which

suggests the use of Schinzel's hypothesis we show that the integral

Brauer-Manin obstruction is the only obstruction to the integral Hasse

principle for an infinite family of generalised affine Châtelet surfaces.

Moreover, we show that the set of integral points on any surface in this

family for which there is no integral Brauer-Manin obstruction satisfies a

strong approximation property away from infinity. We do so by exploiting

the conic bundle structure of such surfaces over the affine line. A

corollary of Schinzel's hypothesis allows us to reduce the main problem to

establishing the integral Hasse principle for an affine conic corresponding

to a specific fibre in the surface. We can handle this with a new idea

involving a class group argument for a quadratic number field associated

with the conic.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/246