We discuss some aspects of the behavior of specialization at a finite place of Néron–Severi groups of K3 surfaces over number fields. We give optimal lower bounds for the Picard number of such specializations, thus answering a question of Elsenhans and Jahnel. As a consequence of these results, we show that it is possible to compute explicitly the Picard number of any given K3 surface over a number field.

We generalize the linear algebra setting of Tate’s central extension to arbitrary dimension. In general, one obtains a Lie $\left(n+1\right)$-cocycle. We compute it to some extent. The construction is based on a Lie algebra variant of Beilinson’s adelic multidimensional residue symbol, generalizing Tate’s approach to the local residue symbol for $1$-forms on curves.

For a binary quadratic form $Q$, we consider the action of ${SO}_Q$ on a 2-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order.

We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant $D$, the shape is equidistributed in the class group ${Cl}_D$ of binary quadratic forms of discriminant $D$. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field.

We explicitly bound the Faltings height of a curve over $\overline{\mathbb{Q}}$ polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings’ delta invariant and the self-intersection of the dualising sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes–Edixhoven–Bruin algorithm to compute coefficients of modular forms for congruence subgroups of ${SL}_2\left(\mathbb{Z}\right)$ runs in polynomial time under the Riemann hypothesis for $\zeta$-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of ${\mathbb{P}}_{\mathbb{Z}}^1$ with fixed branch locus.

We introduce a new invariant for subcategories $\mathcal{X}$ of finitely generated modules over a local ring $R$ which we call the radius of $\mathcal{X}$. We show that if $R$ is a complete intersection and $\mathcal{X}$ is resolving, then finiteness of the radius forces $\mathcal{X}$ to contain only maximal Cohen–Macaulay modules. We also show that the category of maximal Cohen–Macaulay modules has finite radius when $R$ is a Cohen–Macaulay complete local ring with perfect coefficient field. We link the radius to many well-studied notions such as the dimension of the stable category of maximal Cohen–Macaulay modules, finite/countable Cohen–Macaulay representation type and the uniform Auslander condition.

A generalized Bogomolov–Gieseker inequality for tilt-stable complexes on a smooth projective threefold was conjectured by Bayer, Toda, and the author. We show that such inequality holds true in general if it holds true when the polarization is sufficiently small. As an application, we prove it for the three-dimensional projective space.

We construct noncommutative multidimensional versions of overconvergent power series rings and Robba rings. We show that the category of étale $\left(\phi ,\Gamma \right)$-modules over certain completions of these rings is equivalent to the category of étale $\left(\phi ,\Gamma \right)$-modules over classical overconvergent or Robba rings as the case may be (hence also to the category of $p$-adic Galois representations of ${\mathbb{Q}}_p$). In the case of Robba rings, the assumption of étaleness is not necessary, so there exists a notion of trianguline objects in this sense.

Let $H$ be a connected reductive group over an algebraically closed field of characteristic zero, and let $\Gamma$ be an abstract group. In this note, we show that every homomorphism of Grothendieck semirings $\varphi :K_0^{+}\left[H\right]\to K_0^{+}\left[\Gamma \right]$, which maps irreducible representations to irreducible, comes from a group homomorphism $\rho :\Gamma \to H\left(K\right)$. We also connect this result with the Langlands conjectures.