In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Néron–Severi group over an algebraic closure of the base field, is high enough, more structure is known and more can be said. However, until recently not a single explicit K3 surface was known to have geometric Picard number one. We give explicit examples of such surfaces over the rational numbers. This solves an old problem that has been attributed to Mumford. The examples we give also contain infinitely many rational points, thereby answering a question of Swinnerton-Dyer and Poonen.

We show that every fusion system on a $p$-group $S$ is equal to the fusion system associated to a discrete group $G$ with the property that every $p$-subgroup of $G$ is conjugate to a subgroup of $S$.

We construct a two-level weighted topological quantum field theory whose structure coefficients are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local Gromov–Witten theory of curves of Bryan and Pandharipande. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized ${\mathbb{P}}^1$. The Frobenius algebras we obtain are one-parameter deformations of the class algebra of the symmetric group $S_d$. In certain special cases we are able to produce explicit closed formulas for such deformations in terms of the representation theory of $S_d$.

We consider mod $p$ modular Galois representations which are unramified at $p$ such that the Frobenius element at $p$ acts through a scalar matrix. The principal result states that the multiplicity of any such representation is bigger than $1$.

In this paper we will establish functional equations for Mahler measures of families of genus-one two-variable polynomials. These families were previously studied by Beauville, and their Mahler measures were considered by Boyd, Rodriguez Villegas, Bertin, Zagier, and Stienstra. Our functional equations allow us to prove identities between Mahler measures that were conjectured by Boyd. As a corollary, we also establish some new transformations for hypergeometric functions.