For a prime $p$, the OM algorithm finds the $p$-adic factorization of an irreducible polynomial $f\in \mathbb{Z}[x]$ in polynomial time. This may be applied to construct $p$-integral bases in the number field $K$ defined by $f$. In this paper, we adapt the OM techniques to work with a positive integer $N$ instead of $p$. As an application, we obtain an algorithm to compute global integral bases in $K$, which does not require a previous factorization of the discriminant of $f$.

We develop explicit techniques to investigate algebraic quasihyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth’s sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two.

We also improve lower bounds on the dimension of the space of symmetric differentials on surfaces with $A_1$-singularities, and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing $3\times 3$ magic squares of squares are all algebraically quasihyperbolic.

The Olson constant $\mathcal{𝒪}L({\mathbb{𝔽}}_p^d)$ represents the minimum positive integer $t$ with the property that every subset $A\subset {\mathbb{𝔽}}_p^d$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{𝒪}L({\mathbb{𝔽}}_p^d)$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$.

We prove that for any fixed $d\ge 2$ and , the Olson constant of ${\mathbb{𝔽}}_p^d$ satisfies the inequality

$$\mathcal{𝒪}L({\mathbb{𝔽}}_p^d)\le (d-1+\epsilon )p$$

for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

We study the automorphisms of modular curves associated to Cartan subgroups of ${\text{GL}}_2$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for nonsplit curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^{+}(p)$ has no nontrivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one nontrivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^{\ast }(n):=X_0(n)∕W$, where $W$ is the group generated by the Atkin–Lehner involutions of $X_0(n)$ and $n$ is a large enough square.

We study the topological spectrum ${\text{Spec}}_{\text{Top}}(R)$ of a seminormed ring $R$ which we define as the space of prime ideals $\mathfrak{𝔭}$ such that $\mathfrak{𝔭}$ equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring $R$ we show that ${\text{Spec}}_{\text{Top}}(R)$ is a quasicompact sober topological space. When $R$ is a perfectoid Tate ring we construct a natural homeomorphism

$${\text{Spec}}_{\text{Top}}(R)\simeq {\text{Spec}}_{\text{Top}}(R^{♭})$$

between the topological spectrum of $R$ and the topological spectrum of its tilt $R^{♭}$. As an application, we prove that a perfectoid Tate ring $R$ is an integral domain if and only if its tilt is an integral domain.

The space of $n\times m$ complex matrices can be regarded as an algebraic variety on which the group ${\text{GL}}_n\times {\text{GL}}_m$ acts. There is a rich interaction between geometry and representation theory in this example. In an important paper, de Concini, Eisenbud, and Procesi classified the equivariant ideals in the coordinate ring. More recently, we proved a noetherian result for families of equivariant modules as $n$ and $m$ vary. We establish analogs of these results for the space of $(n|n)\times (m|m)$ isomeric matrices with respect to the action of $Q_n\times Q_m$, where $Q_n$ is the automorphism group of the isomeric structure (commonly known as the “queer supergroup”). Our work is motivated by connections to the Brauer category and the theory of twisted commutative algebras.

We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macaulay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regularity.