Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

Poincaré duality isomorphisms in tensor categories

If for a vector space V of dimension g over a characteristic zero field we denote by ${\wedge }^{i}V$ its alternating powers, and by $V^{\vee }$ its linear dual, then there are natural Poincaré isomorphisms:

${\wedge }^i{V}^{\vee }?{\wedge }^{g?i}V.$

We describe an analogous result for objects in rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor categories.     

Le nombre des systèmes locaux ℓ-adiques sur une courbe

Let $X_1$ be a projective, smooth, and geometrically connected curve over ${\mathbb{F}}_q$ with $q=p^n$ elements where p is a prime number, and let X be its base change to an algebraic closure of ${\mathbb{F}}_q$. The goal of this article is to announce a formula for the number of irreducible ?-adic local systems ($?\ne p$) with a fixed rank over X fixed by the Frobenius endomorphism. This number behaves like a Grothendieck–Lefschetz fixed point formula for a variety over ${\mathbb{F}}_q$, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. We also sketch our method, which consists in passing to the automorphic side by Langlands correspondence, then calculating all the terms in Arthur's non-invariant trace formula and linking the geometric part of trace formula to the number of ${\mathbb{F}}_q$-points of the moduli space of stable Higgs bundles.     

On the Schur multiplier of special p-groups

Let G be a special p-group minimally generated by $d\ge 3$ elements and having derived subgroup of order $p^{\frac{1}{2}d(d?1)}$. Berkovich asked to find the Schur multiplier and covering groups of such groups G Berkovich and Janko (2011) [1]. We try to give an answer to this question in this article.     

Club isomorphisms on higher Aronszajn trees

We prove the consistency, assuming an ineffable cardinal, of the statement that CH holds and any two normal countably closed ${\omega }_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham–Shelah [1] that any two normal ${\omega }_1$-Aronszajn trees are club isomorphic, which follows from PFA. The statement that any two normal countably closed ${\omega }_2$-Aronszajn trees are club isomorphic implies that there are no ${\omega }_2$-Suslin trees, so our proof also expands on the method of Laver–Shelah [5] for obtaining the ${\omega }_2$-Suslin hypothesis.     

Local uniformization and arc spaces

Let X be a variety over a perfect field k and let $X_{\infty }$ be its space of arcs. Given a closed subset Z of X, let $X_{\infty }^Z$ denote the subscheme of $X_{\infty }$ consisting of all arcs centered at some point of Z. We prove that Local Uniformization implies that $X_{\infty }^Z$ has a finite number of irreducible components for each closed subset Z of X. In particular, Local Uniformization implies that $X_{\infty }^{\mathit{Sing}X}$ has a finite number of irreducible components.     

On the ordinariness of coverings of stable curves

In the present paper, we study the ordinariness of coverings of stable curves. Let $f:Y\to X$ be a morphism of stable curves over a discrete valuation ring R with algebraically closed residue field of characteristic $p>0$. Write S for Spec R and η (resp. s) for the generic point (resp. closed point) of S. Suppose that the generic fiber $X_{\eta }$ of X is smooth over η, that the morphism $f_{\eta }:Y_{\eta }\to X_{\eta }$ over η on the generic fiber induced by f is a Galois étale covering (hence $Y_{\eta }$ is smooth over η too) whose Galois group is a solvable group G, that the genus of the normalization of each irreducible component of the special fiber $X_s$ is ≥2, and that $Y_s$ is ordinary. Then we have that the morphism $f_s:Y_s\to X_s$ over s induced by f is an admissible covering. This result extends a result of M. Raynaud concerning the ordinariness of coverings to the case where $X_s$ is a stable curve. If, moreover, we suppose that G is a p-group, and that the p-rank of the normalization of each irreducible component of $X_s$ is ≥2, we can give a numerical criterion for the admissibility of $f_s$.     

Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

Embeddings of rank-2 tori in algebraic groups

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type $A_2$, $G_2$ and $F_4$ defined over k, via their rank-2 k-tori. Simple, simply connected groups of type $A_2$ play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of $A_n$ type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type $A_2$, $G_2$ and $F_4$ in terms of the mod-2 Galois cohomological invariants attached with these groups. The results in this paper and our earlier paper ([6]) show that the mod-2 invariants of groups of type $G_2,F_4$ and $A_2$ are controlled by their k-subgroups of type $A_1$ and $A_2$ as well as the unitary k-tori embedded in them.     

Generic expansion and Skolemization in NSOP1 theories

We study expansions of ${\mathrm{NSOP}}_1$ theories that preserve ${\mathrm{NSOP}}_1$. We prove that if T is a model complete ${\mathrm{NSOP}}_1$ theory eliminating the quantifier ${?}^{\infty }$, then the generic expansion of T by arbitrary constant, function, and relation symbols is still ${\mathrm{NSOP}}_1$. We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an ${\mathrm{NSOP}}_1$ theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in ${\mathrm{NSOP}}_1$ theories, adding instances of algebraic independence to their conclusions.     

The classification of Leonard triples of QRacah type

Let $\mathbb{K}$ denote an algebraically closed field. Let V denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis of V with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the Leonard triples of QRacah type up to isomorphism. We show that any Leonard triple of QRacah type satisfies the ${\mathbb{Z}}_3$-symmetric Askey-Wilson relations.     

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equation

$\mathrm{\Delta }u+f(u)=0$

under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\mathrm{\Delta }u+u^p=0$ and $\mathrm{\Delta }u+e^u=0$.