Two-torsion of the Brauer group of an elliptic surface

The Brauer group of a variety encodes important arithmetic information. For instance, it plays an important role in the Brauer-Manin obstruction, which governs the existence of rational points on many varieties. In this thesis we describe the $2$-torsion part of the Brauer group of certain elliptic surfaces. In particular, we compute explicit representatives of these elements in terms of quaternion algebras over the function field of the surface.