We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds’ method.

We describe a conjectural stratification of the Brill–Noether variety for general curves of fixed genus and gonality. As evidence for this conjecture, we show that this Brill–Noether variety has at least as many irreducible components as predicted by the conjecture and that each of these components has the expected dimension. Our proof uses combinatorial and tropical techniques. Specifically, we analyze containment relations between the various strata of tropical Brill–Noether loci identified by Pflueger in his classification of special divisors on chains of loops.

We study the maximal multiplicity locus of a variety X over a field of characteristic $\mathit{p}>0$ that is provided with a finite surjective radicial morphism $\mathit{\delta }:\mathit{X}\to \mathit{V}$, where V is regular, for example, when $\mathit{X}\subset {\mathbb{A}}^{\mathit{n}+1}$ is a hypersurface defined by an equation of the form ${\mathit{T}}^{\mathit{q}}-\mathit{f}({\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}})=0$ and δ is the projection onto $\mathit{V}:=\mathrm{Spec}(\mathit{k}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}}])$. The multiplicity along points of X is bounded by the degree, say d, of the field extension $\mathit{K}(\mathit{V})\subset \mathit{K}(\mathit{X})$. We denote by ${\mathit{F}}_{\mathit{d}}(\mathit{X})\subset \mathit{X}$ the set of points of multiplicity d. Our guiding line is the search for invariants of singularities $\mathit{x}\in {\mathit{F}}_{\mathit{d}}(\mathit{X})$ with a good behavior property under blowups ${\mathit{X}}^{\prime }\to \mathit{X}$ along regular centers included in ${\mathit{F}}_{\mathit{d}}(\mathit{X})$, which we call invariants with the pointwise inequality property.

A finite radicial morphism $\mathit{\delta }:\mathit{X}\to \mathit{V}$ as above will be expressed in terms of an ${\mathcal{O}}_{\mathit{V}}^{\mathit{q}}$-submodule $\mathcal{M}\subseteq {\mathcal{O}}_{\mathit{V}}$. A blowup ${\mathit{X}}^{\prime }\to \mathit{X}$ along a regular equimultiple center included in ${\mathit{F}}_{\mathit{d}}(\mathit{X})$ induces a blowup ${\mathit{V}}^{\prime }\to \mathit{V}$ along a regular center and a finite morphism ${\mathit{\delta }}^{\prime }:{\mathit{X}}^{\prime }\to {\mathit{V}}^{\prime }$. A notion of transform of the ${\mathcal{O}}_{\mathit{V}}^{\mathit{q}}$-module $\mathcal{M}\subset {\mathcal{O}}_{\mathit{V}}$ to an ${\mathcal{O}}_{{\mathit{V}}^{\prime }}^{\mathit{q}}$-module ${\mathcal{M}}^{\prime }\subset {\mathcal{O}}_{{\mathit{V}}^{\prime }}$ will be defined in such a way that ${\mathit{\delta }}^{\prime }:{\mathit{X}}^{\prime }\to {\mathit{V}}^{\prime }$ is the radicial morphism defined by ${\mathcal{M}}^{\prime }$. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on ${\mathcal{O}}_{\mathit{V}}^{\mathit{q}}$-submodules $\mathcal{M}\subset {\mathcal{O}}_{\mathit{V}}$.

In this paper, we study a Hamilton–Jacobi flow ${\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\}}_{\mathit{t}>0}$ starting from the Takagi function τ. The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on $\mathbb{R}$. As the first result of this paper, we derive an explicit representation of $\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\}$. It turns out that ${\mathit{H}}_{\mathit{t}}\mathit{\tau }$ is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that $\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\}$ has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.

Let p be a prime number. We give several results towards a particular instance of a conjecture of Einsiedler and Kleinbock asserting that every p-adic number x satisfies

$$\underset{\mathit{a},\mathit{b}\in \mathbb{Z}\setminus \{0\}}{inf}|\mathit{a}\mathit{b}|·|\mathit{a}\mathit{x}-\mathit{b}{|}_{\mathit{p}}=0.$$

We highlight a close relationship between this conjecture and the (still open) p-adic Littlewood conjecture, according to which every real number ξ satisfies

$$\underset{\mathit{q}\in \mathbb{Z},\mathit{q}\ge 1}{inf}\mathit{q}·\Vert \mathit{q}\mathit{\xi }\Vert ·|\mathit{q}{|}_{\mathit{p}}=0.$$

Furthermore, we discuss the analogues of these conjectures over fields of power series.

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on ${\mathbb{R}}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.

We introduce the notion of an isotropic quantum state associated with a Bohr–Sommerfeld manifold in the context of Berezin–Toeplitz quantization of general prequantized symplectic manifolds, and we study its semiclassical properties using the off-diagonal expansion of the Bergman kernel. We then show how these results extend to the case of noncompact orbifolds and give an application to relative Poincaré series in the theory of automorphic forms.

The paper was withdrawn by the author after its first appearance online, upon learning of a gap in the proof of Theorem 3.13 and an issue with a published result on which Theorem 3.2 was based. A complete proof of Theorem 3.13, with slightly more restrictive hypotheses, can now be found in: Marco Boggi, “Notes on hyperelliptic mapping class groups”, arXiv:2110.13534. A revised version of this preprint will include all of the correct results in the retracted paper.