Obstructions to integral points on affine Ch\^atelet surfaces

We consider the Brauer-Manin obstruction to the existence of integral points on affine surfaces defined by $x^2 - ay^2 = P(t)$ over a number field. We enumerate the possibilities for the Brauer groups of certain families of such surfaces, and show that in contrast to their smooth compactifications, the Brauer groups of these affine varieties need not be generated by cyclic (e.g. quaternion) algebras. Concrete examples are given of such surfaces over $\mathbb{Q}$ which have solutions in $\mathbb{Z}_{p}$ for all $p$ and solutions in $\mathbb{Q}$, but for which the failure of the integral Hasse principle cannot be explained by a Brauer-Manin obstruction. The methods of this paper build on the ideas of several recent papers in the literature: we study the Brauer groups (modulo constant algebras) of affine Ch\^atelet surfaces with a focus on explicitly representing, by central simple algebras over the function field of $X$, a finite set of classes of the Brauer group which generate the quotient. Notably, we develop techniques for constructing explicit representatives of non-cyclic Brauer classes on affine surfaces, as well as provide an effective algorithm for the computation of local invariants of these Brauer classes via effective lifting of cocycles in Galois cohomology.