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For two varieties of dimension $5$ constructed as double covers of ${\mathbbP}^5$ branched along the union of $12$ hyperplanes, we prove that the number ofpoints 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 knownin dimension $\le 3$, but very few in higher dimension. In addition, we use anidea of Burek to construct quotients of our varieties for which the pointcounts mod $p$ are expressible in terms of Artin symbols and the coefficientsof a single modular form of weight $6$.

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