We show that, up to isomorphism, the number of $Q$-points is either finite, $2^{\mathfrak{d}}$ or $2^{\mathfrak{c}}$. This answers a question asked by Borodulin-Nadzieja, Mart\'{i}nez-Celis, Morawski and \'Swierczy\'nska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many $Q$-points implies the existence of non-atomic $Q$-measures, and of $2^{\mathfrak{c}}$-many Tukey-top $Q$-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..
Let $\Pi$ be a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{3}(\mathbb{A}_{\mathbb{Q}})$ of weight $(a, 0, -a)$, and let $p$ be a prime. Generalising work of Loeffler--Williams in the nearly-ordinary case, we construct $p$-adic $L$-functions for each small slope regular $P_{1}$-refinement of $...
It was known at least to L.E. Dickson in 1901 that $\mathrm{SL}(2,q)$, in its natural action on $\mathbb{F}_q^2\setminus\{0\}$, has a sharply transitive subgroup only when $q\in\{2,3,5,7,11\}$. For $q$ prime, this result stems from Galois'letter to Chevalier in 1832. We extend this result to sharply transitive subsets...
In this note, we prove that an arbitrary product of $W$-spaces is a $\kappa$-Fr\'{e}chet-Urysohn space. We also show that the boundary tightness of a product $X \times Y$ is at most $\kappa$ provided that $X$ is compact with tightness $t(X) \leq \kappa$ and $Y$ has boundary tightness $tb(Y) \leq \kappa$. The first resu...
Let $P_q$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_q]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the {\it hit Peterson problem} of finding a minimal set of generators for $P_q$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$...
Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Kl\'asz...
Let $H_{27}=\mathrm{UT}_3(\mathbb{F}_3)$ be the nonabelian group of order $27$ and exponent $3$. We prove that $\mathsf{d}(H_{27}\times C_3^r)=2r+6$ for every integer $r\geq0$. The proof combines an affine coefficient identity in the group algebra of an elementary abelian group with a decomposition of the nonorthogonal...
André Volkmann· 0 citations
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