Noninvariance of the BrauerManin obstruction for surfaces
Abstract
In this paper, we study the properties of weak approximation with BrauerManin obstruction and the Hasse principle with BrauerManin obstruction for surfaces with respect to field extensions of number fields. We assume a conjecture of M. Stoll. For any nontrivial extension of number fields $L/K,$ we construct two kinds of smooth, projective, and geometrically connected surfaces defined over $K.$ For the surface of the first kind, it has a $K$rational point, and satisfies weak approximation with BrauerManin obstruction off $\infty_K,$ while its base change by $L$ does not so off $\infty_L.$ For the surface of the second kind, it is a counterexample to the Hasse principle explained by the BrauerManin obstruction, while the failure of the Hasse principle of its base change by $L$ cannot be so. We illustrate these constructions with explicit unconditional examples.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 arXiv:
 arXiv:2103.01784
 Bibcode:
 2021arXiv210301784W
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 arXiv admin note: substantial text overlap with arXiv:2010.04919