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We give a geometric interpretation (and we deduce an explicit formula) for two types of exponential sums, one of which is the third moment of Kloosterman sums over ${\mathbf{F}}_q$ of type $K({\nu }^2;q)$. We establish a connection between the sums considered and the number of ${\mathbf{F}}_q$-rational points on explicit smooth projective surfaces, one of which is a $K3$ surface, whereas the other is a smooth cubic surface. As a consequence, we obtain, applying Grothendieck-Lefschetz theory, a generalized formula for the third moment of Kloosterman...

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group $\mathbf{G}$ over $\mathbf{Q}$ typically have splitting field with Galois group isomorphic to the Weyl group of $\mathbf{G}$, we construct an explicit monic integral polynomial of degree $240$ whose splitting field has Galois group the Weyl group of the exceptional group of type ${\mathbf{E}}_8$.