Brauer groups for diagonal surfaces

Damián Gvirtz

In joint work with A. Skorobogatov, we propose a framework to determine the Brauer group of a projective diagonal surface \(X\) over a number field \(k\). Our approach utilises results for the complex cohomology of Fermat varieties (Pham, Looijenga) and their associated Galois representations (Weil, Katz, Shioda, Ulmer). For \(k=Q\) or \(\mathbf Q(i)\), we classify the Brauer group for all diagonal quartic surfaces with rational coefficients.