We describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

This paper contains two results on Hodge loci in $\mathsf{M}_{g}$. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in $\mathsf{M}_{g}$. It is proved that the image under the period map of a divisor in $\mathsf{M}_{g}$ is not contained in a proper totally geodesic subvariety of $\mathsf{A}_{g}$. It follows that a Hodge locus in $\mathsf{M}_{g}$ has codimension at least 2.

We consider systems of quasi-periodic linear differential equations associated to a ‘resonant’ frequency vector ${\it\omega}$, namely, a vector whose coordinates are not linearly independent over $\mathbb{Z}$. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a ‘resonant cocycle’. We also apply our results to the non-resonant case: we obtain sufficient conditions for reducibility.

It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Françoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa–Holm system and the (non-periodic) Calogero–Françoise system in terms of theta functions.

AMS 2000 Mathematics subject classification: Primary 35Q51; 37J35; 35Q53

We construct a weak Hilbert space that is a twisted Hilbert space.

Nous démontrons des théorèmes d'immersion holomorphe dans un espace projectif complexe pour des variétés kählériennes non compactes et des laminations par variétés complexes qui admettent un fibré en droites holomorphe strictement positif. En particulier, nous montrons que sur une lamination compacte par surfaces de Riemann, les fonctions méromorphes séparent les points si et seulement si aucun cycle feuilleté n'est homologue à $0$.

We prove holomorphic immersion theorems in a finite dimensional complex projective space for kählerian non-compact manifolds and for laminations by complex manifolds that carry a line bundle of positive curvature. In particular, we prove that on a Riemann surfaces lamination of a compact space, the space of meromorphic functions separates points if and only if every foliation cycle is non homologous to $0$.

Let $\Gamma $ be a finite set, and $X\ni x$ a fixed kawamata log terminal germ. For any lc germ $(X\ni x,B:=\sum _{i} b_iB_i)$, such that $b_i\in \Gamma $, Nakamura’s conjecture, which is equivalent to the ascending chain condition conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor E over $X\ni x$, such that $a(E,X,B)=\mathrm {mld}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura’s conjecture to the setting that $X\ni x$ is not necessarily fixed and $\Gamma $ satisfies the descending chain condition, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such E.

Nous considérons l’intersection d’une sous-variété $X$ d’une variété abélienne $A$ avec l’union de tous les sous-groupes de $A$ obtenus comme somme d’un sous-groupe donné $\varGamma$ de rang fini et d’un sous-groupe algébrique de dimension donnée. Nous montrons que, quitte à retirer à $X$ un certain ensemble exceptionnel, l’intersection est de hauteur bornée. L’énoncé est optimal pour une courbe $X$. Nous étudions également les épaississements $\varGamma_{\varepsilon}$ introduits par Poonen. La démonstration repose sur une généralisation uniforme de la méthode de Vojta et sur des calculs de nombres d’intersection de cycles réels sur $A$.

We study the intersection of a subvariety $X$ of an abelian variety $A$ over $\bar{\mathbb{Q}}$ with the union of all the subgroups obtained as a sum of a given finite rank subgroup $\varGamma$ and an algebraic subgroup of $A$ of given dimension $d$. Our main result asserts that if we remove a suitable exceptional subset from $X$ then the intersection is a set of bounded height. In some cases, this combines with the output of Part I to yield finiteness. In terms of boundedness of the height, we get an optimal statement for a curve $X$ with $d=2$. We also deal with the fattenings $\varGamma_\varepsilon$ introduced by Poonen. The proof rests on a suitably uniform generalization of the method of Vojta and on computations of intersection numbers of real cycles on $A$.

Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

We present a construction that enables one to find Banach spaces $X$ whose sets $\operatorname{NA}(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math. (to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal. 272 (2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $\operatorname{NA}(X)$ for the original norm is not “too large”. The construction can be applied to every Banach space containing $c_{0}$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_{0}$. We also provide some geometric properties of the norms we have constructed.

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.

In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are:

$$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$

$n,k>0$

$S_{k}$

$k$

$S_{k}^{n}$

$n$

$S_{k}$

$m(G^{n})$

$n\rightarrow \infty$

$G$

Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$. We prove that every $R$-homomorphism ${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$ is essentially the evaluation homomorphism at a certain point $p\in F^{n}$ adjacent to the extended semialgebraic set $M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when $M$ is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent:

1. (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ;

2. (ii) Every operator from E into F attains its norm;

3. (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ ,

where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $\mathbb{R}^{n+1}$, for $n\geq3$, which generalize the well known two-dimensional ‘Riemann minimal surfaces’. The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the two-dimensional case, they are not foliated by spheres.

Résumé Nous prouvons l’existence d’une famille à un paramètre d’hypersurfaces de $\mathbb{R}^{n+1}$, pour $n\geq 3$, qui sont minimales et qui généralisent les surfaces minimales de Riemann. Les hypersurfaces que nous obtenons sont des hypersurfaces complètes, simplement périodiques et qui ont une infinité de bouts hyperplans parallèles. Contrairement au cas des surfaces, i.e. $n=2$, ces hypersurfaces ne sont pas feuilletées par des sphères.

We introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

We construct some cusped finite-volume hyperbolic n-manifolds $M^n$ that fibre algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi _1(M^n) \to {\mathbb {Z}}$ with finitely generated kernel. The kernel is also finitely presented in the dimensions $n=7, 8$, and this leads to the first examples of hyperbolic n-manifolds $\widetilde M^n$ whose fundamental group is finitely presented but not of finite type. These n-manifolds $\widetilde M^n$ have infinitely many cusps of maximal rank and, hence, infinite Betti number $b_{n-1}$. They cover the finite-volume manifold $M^n$. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P^5, \ldots , P^8$, and then applying some arguments of Jankiewicz, Norin and Wise [18] and Bestvina and Brady [7]. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P^n$, and the algebra of integral octonions for the crucial dimensions $n=7,8$.

We prove that n-hypergraphs can be interpreted in e-free perfect PAC fields in particular in pseudofinite fields. We use methods of function field arithmetic, more precisely we construct generic polynomials with alternating groups as Galois groups over a function field.

In this note, we prove that the Drinfeld–Grinberg–Kazhdan theorem on the structure of formal neighborhoods of arc schemes at a nonsingular arc does not extend to the case of singular arcs.

Let $p:X\rightarrow Y$ be an algebraic fiber space, and let $L$ be a line bundle on $X$. In this article, we obtain a curvature formula for the higher direct images of $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ restricted to a suitable Zariski open subset of $X$. Our results are particularly meaningful if $L$ is semi-negatively curved on $X$ and strictly negative or trivial on smooth fibers of $p$. Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

We study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.