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Integral points on generalised affine Châtelet surfaces

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Vladimir Mitankin
Universidade Federal do Rio de Janeiro
Mit, 2018-01-10 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

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.

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