We construct a finite subgroup of Brauer–Manin obstruction for detecting the existence of integral points on integral models of principle homogeneous spaces of multi‐norm tori. Several explicit examples are provided.

Results on L-functions of Artin-Schreier coverings by Dwork, Bombieri and Adolphson-Sperber are generalized to L-functions of Witt coverings.

Let K /k be an extension of number fields, and let P(t) be a quadratic polynomial over k. Let X be the affine variety defined by P(t) = N K /k (z). We study the Hasse principle and weak approximation for X in three cases. For [K : k] = 4 and P(t) irreducible over k and split in K , we prove the Hasse principle and weak approximation. For k = Q with arbitrary K , we show that the BrauerManin obstruction to the Hasse principle and weak approximation is the only one. For [K : k] = 4 and P(t) irreducible over k, we determine the Brauer group of smooth proper models of X . In a case where it is non-trivial, we exhibit a counterexample to weak approximation.

Ghosh and Sarnak have studied integral points on surfaces defined by an equation x 2 + y 2 + z 2 − xyz = m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m = 0, 4, the Brauer group does not control strong approximation.