2-adic point counting on $K3$ surfaces

This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the $2$-adic orthogonal group. Combining the new approach with a $p$-adic method, we count the number of points on some $K3$ surfaces over the field ${\text{bbF}}_{\text{\hspace{0.17em}⁣}p}$, for all primes $p<{10}^8$.