K3 surfaces are the two-dimensional analogue of elliptic curves, and their arithmetic is governed by how the absolute Galois group acts on the 22-dimensional second cohomology, split into the algebraic Neron-Severi lattice and its transcendental complement. A student learns why Picard number 20 (singular K3 surfaces) forces a two-dimensional Galois representation and hence modularity by a weight-3 CM newform, how point counting over finite fields plus the Lefschetz fixed point formula and the Artin-Tate conjecture bound the Picard number (van Luijk's method), and what is known about density and potential density of rational points on K3 surfaces.
This arXiv paper by Matthias Schutt (posted September 2008) covers arithmetic of K3 surfaces, important higher-dimensional varieties in arithmetic algebraic geometry. The paper reviews recent develop…