Modularity of two double covers of ${\mathbb P}^5$ branched along $12$ hyperplanes

Abstract

For two varieties of dimension $5$ constructed as double covers of ${\mathbb P}^5$ branched along the union of $12$ hyperplanes, we prove that the number of points over ${\mathbb F}_p$ can be expressed in terms of Artin symbols and the $p$th Fourier coefficients of modular forms. Many analogous results are known in dimension $\le 3$, but very few in higher dimension. In addition, we use an idea of Burek to construct quotients of our varieties for which the point counts mod $p$ are expressible in terms of Artin symbols and the coefficients of a single modular form of weight $6$.