Let $p$ be a prime number and $\xi$ an irrational $p$-adic number. Its irrationality exponent $\mu \left(\xi \right)$ is the supremum of the real numbers $\mu$ for which the system of inequalities

has a solution in integers $x$, $y$ for arbitrarily large real number $X$. Its multiplicative irrationality exponent ${\mu }^{\times }\left(\xi \right)$ (resp., uniform multiplicative irrationality exponent ${\hat{\mu }}^{\times }\left(\xi \right))$ is the supremum of the real numbers $\hat{\mu }$ for which the system of inequalities

has a solution in integers $x$, $y$ for arbitrarily large (resp., for every sufficiently large) real number $X$. It is not difficult to show that $\mu \left(\xi \right)\le {\mu }^{\times }\left(\xi \right)\le 2\mu \left(\xi \right)$ and ${\hat{\mu }}^{\times }\left(\xi \right)\le 4$. We establish that the ratio between the multiplicative irrationality exponent ${\mu }^{\times }$ and the irrationality exponent $\mu$ can take any given value in $[1, 2]$. Furthermore, we prove that ${\hat{\mu }}^{\times }\left(\xi \right)\le (5+\sqrt{5})/2$ for every $p$-adic number $\xi$.

Let $A$ be an abelian variety defined over a number field and of dimension $g$. When $g\le 2$, by the recent work of Sawin, we know the exact (nonzero) value of the density of the set of primes which are ordinary for $A$. In higher dimension very little is known. We show that if $g=3$ and $A$ has multiplication by an imaginary quadratic field $E$, then there exists a nonzero density set of ordinary primes for $A$. We reach the same conclusion if $g=4$ and the pair $(A,E)$ has signature $(2,2)$. We also obtain partial results when $g=3$ and $A$ has multiplication by a totally real cubic field. We show that our methods also apply to certain abelian varieties of Albert type IV of higher dimension. These results are derived from an extended version of the $\ell$-adic methods of Katz, Ogus, and Serre in the presence of extra endomorphisms.

Following Bertolini and Darmon’s method, with “Ihara’s lemma” among other conditions Longo and Wang proved one divisibility of the Iwasawa main conjecture for Hilbert modular forms of weight $2$ and general low even parallel weight in the anticyclotomic setting respectively. In this paper, we remove the “Ihara’s lemma” condition in their results.

Let $p$ be a prime number and $q=p^m$, with $m\ge 1$ if $p\ne 2,3$ and otherwise. Let $\mathrm{\Omega }$ be any non-trivial twist for the complex group algebra of ${\mathbf{PSL}}_2\left(q\right)$ arising from a $2$-cocycle on an abelian subgroup of ${\mathbf{PSL}}_2\left(q\right)$. We show that the twisted Hopf algebra $\left(\mathrm{ℂ}{\mathbf{PSL}}_2\right(q){)}_{\mathrm{\Omega }}$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki groups, and for ${\mathbf{SL}}_3\left(p\right)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type.

As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group $G$ there is a twist $\mathrm{\Omega }$ for $\mathrm{ℂ}G$, arising from a $2$-cocycle on an abelian subgroup of $G$, such that $(\mathrm{ℂ}G{)}_{\mathrm{\Omega }}$ does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.

We provide examples of vector fields on $({\mathrm{ℂ}}^3,0)$ admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei–Moussu.

We use Lurie’s symmetric monoidal envelope functor to give two new descriptions of $\mathrm{\infty }$-operads: as certain symmetric monoidal $\mathrm{\infty }$-categories whose underlying symmetric monoidal $\mathrm{\infty }$-groupoids are free, and as certain symmetric monoidal $\mathrm{\infty }$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\mathrm{\infty }$-operads, as a localization of a presheaf $\mathrm{\infty }$-category, and we use this to give a simple proof of the equivalence between Lurie’s and Barwick’s models for $\mathrm{\infty }$-operads.

We study period integrals involving Whittaker functions associated to generic irreducible Casselman–Wallach representations of ${GL}_n\left(F\right)$, where $F$ is an archimedean local field. Via the archimedean theory of newforms for ${GL}_n$ developed by the first author, we prove that newforms are weak test vectors for several period integrals, including the ${GL}_n\times {GL}_n$ Rankin–Selberg integral, the Flicker integral, and the Bump–Friedberg integral. By taking special values of these period integrals, we deduce that newforms are weak test vectors for Rankin–Selberg periods, Flicker–Rallis periods, and Friedberg–Jacquet periods. These results parallel analogous results in the nonarchimedean setting proved by the second author, which use the nonarchimedean theory of newforms for ${GL}_n$ developed by Jacquet, Piatetski-Shapiro, and Shalika. By combining these archimedean and nonarchimedean results, we prove the existence of weak test vectors for certain global period integrals of automorphic forms.

We unite elements of category theory, K-theory, and geometric group theory, by defining a class of groups called $k$-cube groups, which act freely and transitively on the product of $k$ trees, for arbitrary $k$. The quotient of this action on the product of trees defines a $k$-dimensional cube complex, which induces a higher-rank graph. We make deductions about the K-theory of the corresponding rank-$k$ graph $C^{\star }$-algebras, and give examples of $k$-cube groups and their K-theory. These are among the first explicit computations of K-theory for an infinite family of rank-$k$ graphs for $k\ge 3$, which is not a direct consequence of the Künneth theorem for tensor products.

We study metric and cohomological properties of Oeljeklaus–Toma manifolds. In particular, we describe the structure of the double complex of differential forms and its Bott–Chern cohomology and we characterize the existence of pluriclosed (aka SKT) metrics in number-theoretic and cohomological terms. Moreover, we prove that they do not admit any Hermitian metric $\omega$ such that $\partial \overline{\partial }{\omega }^k=0$, for $2\le k\le n-2$, and we give explicit formulas for the Dolbeault cohomology of Oeljeklaus–Toma manifolds admitting pluriclosed metrics.

One of the results in our article which appeared in Publ. Mat. 65(2) (2021), 499–528, is that the structure monoid $M(X,r)$ of a left non-degenerate solution $(X,r)$ of the Yang–Baxter equation is a left semi-truss, in the sense of Brzeziński, with an additive structure monoid that is close to being a normal semigroup. Let $\eta$ denote the least left cancellative congruence on the additive monoid $M(X,r)$. It is then shown that $\eta$ is also a congruence on the multiplicative monoid $M(X,r)$ and that the left cancellative epimorphic image $\overline{M}=M(X,r)/\eta$ inherits a semi-truss structure and thus one obtains a natural left non-degenerate solution of the Yang–Baxter equation on $\overline{M}$. Moreover, it restricts to the original solution $r$ for some interesting classes, in particular if $(X,r)$ is irretractable. The proof contains a gap. In the first part of the paper we correct this mistake by introducing a new left cancellative congruence $\mu$ on the additive monoid $M(X,r)$ and show that it also yields a left cancellative congruence on the multiplicative monoid $M(X,r)$, and we obtain a semi-truss structure on $M(X,r)/\mu$ that also yields a natural left non-degenerate solution.

In the second part of the paper we start from the least left cancellative congruence $\nu$ on the multiplicative monoid $M(X,r)$ and show that it is also a congruence on the additive monoid $M(X,r)$ in the case where $r$ is bijective. If, furthermore, $r$ is left and right non-degenerate and bijective, then $\nu =\eta$, the least left cancellative congruence on the additive monoid $M(X,r)$, extending an earlier result of Jespers, Kubat, and Van Antwerpen to the infinite case.

We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The constructive nature of the algorithm gives additional information about the almost disjoint structure of sparse collections.

As applications, we give three results for collections of axis-parallel rectangles in every dimension. The first is a constructive proof of the equivalence between Carleson and sparse collections, first shown by Hänninen. The second is a structure theorem proving that every finite collection $\epsilon$ can be partitioned into $\mathcal{O}\left(N\right)$ sparse subfamilies, where $N$ is the Carleson constant of $\epsilon$. We also give examples showing that such a decomposition is impossible when the geometric assumptions are dropped. The third application is a characterization of the Carleson constant involving only $L^{1,\infty }$ estimates.

We collect some peculiarities of higher-order fractional Laplacians $(-\mathrm{\Delta }{)}^s$, , with special attention to the range $s\in (1,2)$, which show their oscillatory nature. These include the failure of the polarization and Pólya–Szegő inequalities and the explicit example of a domain with sign-changing first eigenfunction. In spite of these fluctuating behaviours, we prove how the Faber–Krahn inequality still holds for any in dimension one.

We study the $p$-tuples of holomorphic vector fields $(X_1,X_2,\dots ,X_p)$ satisfying the barycentric property $\sum _k\text{exp }t{X}_k=p·id$, where $\text{exp }tX$ denotes the flow of $X$.