Recent zbMATH articles in MSC 20Ghttps://zbmath.org/atom/cc/20G2024-06-14T15:52:26.737412ZWerkzeugCounting abelian varieties over finite fields via Frobenius densitieshttps://zbmath.org/1534.110892024-06-14T15:52:26.737412Z"Achter, Jeffrey D."https://zbmath.org/authors/?q=ai:achter.jeffrey-d"Altuğ, S. Ali"https://zbmath.org/authors/?q=ai:altug.salim-ali"Garcia, Luis"https://zbmath.org/authors/?q=ai:garcia.luis-angel|fuentes-garcia.luis|garcia.luis-david|garcia.luis-p-f|garcia.luis-e|garcia.luis-alberto|garcia.luis-pedro"Gordon, Julia"https://zbmath.org/authors/?q=ai:gordon.juliaLet \([X,\lambda]\) be a principally polarized \(g\)-dimensional abelian variety over the finite field \(\mathbb F_q\) where \(q=p^e\) and \(p\) is prime. Let \(I([X,\lambda],q)\) denote its isogeny class, which contains finitely many varieties, and let \(\widetilde\#I([X,\lambda],q)\) be the cardinality weighted by automorphism group. It is proved in the paper under review an Euler-product formula of \(\widetilde\#I([X,\lambda],q)\), which is stated below.
Let \(f(T)\) be the characteristic polynomial of Frobenius of \(X/\mathbb F_q\), let \(\mathrm{GSp}_{2g}\) be the group of symplectic similitudes of a symplectic space of dimension \(2g\), and let \(\mathbb{A}_{\mathrm{GSp}_{2g}}\) be the space of characteristic polynomials of these similitudes. Given a rational prime \(\ell \nmid p \mathrm{disc}(f)\), let
\[
v_\ell := \lim_{n\to\infty} \frac{ \#\lbrace\gamma \in \mathrm{GSp}_{2g}(\mathbb Z_\ell/\ell^n): \mathrm{charpoly}_\gamma(T) = f(T)\ \mathrm{mod }\ \ell^n\rbrace } { \#\mathrm{GSp}_{2g}(\mathbb Z_\ell/\ell^n)/\#\mathbb{A}_{\mathrm{GSp}_{2g}}(\mathbb Z_\ell/\ell^n) }.
\]
We refer to the paper for the definition of \(v_\ell\), where \(\ell \mid p\, \mathrm{disc}(f)\, \infty\), which is more involved. The authors prove the following theorem: Suppose that either \(X\) is ordinary or \(q=p\). Then,
\[
\widetilde\#I([X,\lambda],q) = q^{\frac12 \mathrm{dim}(\mathcal A_g)} \tau_T v_\infty \prod_{\ell} v_\ell,
\]
where \(\tau_T\) is the Tamagawa number of the algebraic torus \(T\) associated with \([X,\lambda]\).
It was proved in [\textit{E.-U. Gekeler}, Int. Math. Res. Not. 2003, No. 37, 1999--2018 (2003; Zbl 1104.11033)] the above theorem for an ordinary elliptic curve, and the work of the paper under review is a generalization of [loc. cit.]. As demonstrated in [\textit{J. D. Achter} and \textit{J. Gordon}, Pac. J. Math. 286, No. 1, 1--24 (2017; Zbl 1379.11065)] for elliptic curves, the size of the isogeny class for the abelian variety is equal to the product of the global volume of a certain adelic quotient and an orbital integral on \(\mathrm{GSp}_{2g}\), and this can be proved using Kottwitz's formula, which is a generalization of Langland's formula for counting points on modular curves over finite fields. The contribution of the paper under review is to decompose the product of the global volume and the integral into an Euler product as described in the theorem. The cohomological and combinatorical aspects of the symplectic similitude groups are much more involved than general linear groups, and little is known about the Tamagawa number \(\tau_T\) to the authors. Available in the paper is the appendix on the Tamagawa number, written by two contributors Thomas Rüd and Wen-Wei Li.
Reviewer: Sungkon Chang (Savannah)Pathologies of the group of R-equivalence classes of a linear algebraic groupshttps://zbmath.org/1534.140452024-06-14T15:52:26.737412Z"Scavia, Federico"https://zbmath.org/authors/?q=ai:scavia.federicoSummary: Let \(k_0\) be a field of characteristic \(p > 0\) and \(k = k_0(t)\), where \(t\) is transcendental over \(k_0\). We give an example of a smooth connected unipotent \(k\)-group \(G\) such that \(G(F)/R\) is non-commutative for some finite separable field extension \(F\) containing \(k\).Existence of embeddings of smooth varieties into linear algebraic groupshttps://zbmath.org/1534.140492024-06-14T15:52:26.737412Z"Feller, Peter"https://zbmath.org/authors/?q=ai:feller.peter"van Santen, Immanuel"https://zbmath.org/authors/?q=ai:stampfli.immanuelThere are three main theorems in the article concerning embedding (closed) of affine varieties (= closed subvarieties of the affine space $\mathbb{A}^{n}(\mathbf{k})$ over an algebraically closed field) into algebraic groups (= closed subgroups of the general linear groups $\mathrm{GL}_{k}(\mathbf{k})$ for some positive integer $k$):
1.\textbf{ Existence Theorem.} Let $G$ be a simple (= has no non-trivial connected normal subgroups) algebraic group and $Z$ a smooth affine variety. If $\dim G \geq 2\dim Z +2 ,$ then $Z$ admits an embedding into $G$.
2. \textbf{Optimality Theorem for even dimensions.} Let $G$ be an algebraic group of dimension $n \geq 1.$ Then, for every integer $d \geq n/2$ there exists a smooth irreducible affine variety $Z$ of dimension $d$ that does not admit an embedding into $G$.
3. \textbf{Surjectivity for rational homology.} Let\textbf{}$f :X \rightarrow Y$ be a proper surjective morphism between complex $n$-dimensional smooth varieties. Then the induced map on $k$-th rational homology groups $H_{k}(X ;\mathbb{Q}) \rightarrow H_{k}(Y ;\mathbb{Q})$ is a surjection for all integers $k \geq 0$.
Reviewer's remark: The last theorem is not new (see [\textit{R. O. Wells jun.}, Pac. J. Math. 53, 281--300 (1974; Zbl 0261.32005)]).
Reviewer: Tadeusz Krasiński (Łódź)A note on the Formanek Weingarten functionhttps://zbmath.org/1534.200102024-06-14T15:52:26.737412Z"Procesi, Claudio"https://zbmath.org/authors/?q=ai:procesi.claudioSummary: The aim of this note is to compare work of \textit{E. Formanek} [J. Algebra 109, 93--114 (1987; Zbl 0625.16015)] on a certain construction of central polynomials with that of \textit{B. Collins} [Int. Math. Res. Not. 2003, No. 17, 953--982 (2003; Zbl 1049.60091)] on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls Weingarten function. By joining these two approaches we succeed in giving a simplified and very natural presentation of both Formanek and Collins's Theory.On chief factors of parabolic maximal subgroups of the group \({}^2F_4(2^{2n+1})\)https://zbmath.org/1534.200152024-06-14T15:52:26.737412Z"Korableva, V. V."https://zbmath.org/authors/?q=ai:korableva.vera-vladimirovnaSummary: This study continues the author's previous papers where a refined description of the chief factors of a parabolic maximal subgroup involved in its unipotent radical was obtained for all (normal and twisted) finite simple groups of Lie type except for the groups \({}^2F_4(2^{2n+1})\) and \(B_l(2^n)\). In present paper, such a description is given for the group \({}^2F_4(2^{2n+1})\). We prove a theorem in which, for every parabolic maximal subgroup of \({}^2F_4(2^{2n+1})\), a fragment of the chief series involved in the unipotent radical of this subgroup is given. Generators of the corresponding chief factors are presented in a table.A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroupshttps://zbmath.org/1534.200632024-06-14T15:52:26.737412Z"Marko, František"https://zbmath.org/authors/?q=ai:marko.frantisekSummary: For a general linear supergroup \(G = \mathrm{GL}(m|n),\) we consider a natural isomorphism \(\phi: G \to U^- \times G_{ev} \times U^+,\) where \textit{\(G_{ev} \)} is the even subsupergroup of \(G\), and \(U^{-,}U^+\) are appropriate odd unipotent subsupergroups of \(G\). We compute the action of odd superderivations on the images \(\phi^{\ast}(x_{ij})\) of the generators of \(K[G]\) extending results established in
[the author, J. Pure Appl. Algebra 219, No. 4, 978--1007 (2015; Zbl 1305.15023); J. Algebra 494, 92--110 (2018; Zbl 1395.17051).
We describe a specific ordering of the dominant weights \(X(T)^+\) of \(\mathrm{GL}(m|n)\) for which there exists a Donkin-Koppinen filtration of the coordinate algebra \(K[G]\). Let \(\Gamma\) be a finitely generated ideal \(\Gamma\) of \(X(T)^+\) and \(O_{\Gamma}(K[G])\) be the largest \(\Gamma\)-subsupermodule of \(K[G]\) having simple composition factors of highest weights \(\lambda \in \Gamma\). We apply combinatorial techniques, using generalized bideterminants, to determine a basis of \(G\)-superbimodules appearing in Donkin-Koppinen filtration of \(O_{\Gamma}(K[G])\) considered initially in
[the author and \textit{A. N. Zubkov}, Transform. Groups 28, No. 2, 911--949 (2023; Zbl 1516.20112)].Minimal parabolic \(k\)-subgroups acting on symmetric \(k\)-varieties corresponding to \(k\)-split groupshttps://zbmath.org/1534.200642024-06-14T15:52:26.737412Z"Hunnell, Mark"https://zbmath.org/authors/?q=ai:hunnell.markSummary: Symmetric \(k\)-varieties are a natural generalization of symmetric spaces to general fields \(k\). We study the action of minimal parabolic \(k\)-subgroups on symmetric \(k\)-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of \(k\)-split groups that depends only on the dimension of a maximal \(k\)-split torus contained within the fixed point group of the involution defining the symmetric \(k\)-variety.Elementary nets (carpets) over a discrete valuation ringhttps://zbmath.org/1534.200652024-06-14T15:52:26.737412Z"Koibaev, Vladimir A."https://zbmath.org/authors/?q=ai:koibaev.v-aSummary: Elementary net (carpet) \( \sigma = (\sigma_{ij})\) is called closed (admissible) if the elementary net (carpet) group \(E(\sigma)\) does not contain a new elementary transvections. The work is related to the question of V. M. Levchuk 15.46 from the Kourovka notebook (closedness (admissibility) of the elementary net (carpet) over a field). Let \(R\) be a discrete valuation ring, \(K\) be the field of fractions of \(R\), \(\sigma = (\sigma_{ij})\) be an elementary net of order \(n\) over \(R\), \(\omega=(\omega_{ij})\) be a derivative net for \(\sigma \), and \(\omega_{ij}\) is ideals of the ring \(R\). It is proved that if \(K\) is a field of odd characteristic, then for the closedness (admissibility) of the net \(\sigma \), the closedness (admissibility) of each pair \((\sigma_{ij}, \sigma_{ji})\) is sufficient for all \(i\neq j\).On the replacement property for \(\mathrm{PSL}(2,p)\)https://zbmath.org/1534.200662024-06-14T15:52:26.737412Z"Cueto Noval, David"https://zbmath.org/authors/?q=ai:cueto-noval.david"Lorenz, Aidan A."https://zbmath.org/authors/?q=ai:lorenz.aidan-a"Zadeoğlu, Baran"https://zbmath.org/authors/?q=ai:zadeoglu.baranSummary: The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups \(\mathrm{PSL}(2, p)\) where \(p\) is a prime larger than 5, first partial results concerning the replacement property were published by
\textit{B. Nachman} [J. Group Theory 17, No. 6, 925--945 (2014; Zbl 1308.20044)]. Second partial results were published by
\textit{H. P. G Lam} [``The Replacement property of \(\mathrm{PSL}(2,p)\) and \(\mathrm{PSL}(2,p^2)\)'', Preprint, \url{arXiv:1709.08745}]. The main goal of this paper is to provide a complete answer for \(\mathrm{PSL}(2, p)\).\(p\)-power conjugacy classes in \(U(n,q)\) and \(T(n,q)\)https://zbmath.org/1534.200672024-06-14T15:52:26.737412Z"Dolfi, Silvio"https://zbmath.org/authors/?q=ai:dolfi.silvio"Singh, Anupam"https://zbmath.org/authors/?q=ai:singh.anupam.1|singh.anupam"Yadav, Manoj K."https://zbmath.org/authors/?q=ai:yadav.manoj-kumarSummary: Let \(q\) be a \(p\)-power where \(p\) is a fixed prime. In this paper, we look at the \(p\)-power maps on unitriangular group \(U(n,q)\) and triangular group \(T(n,q)\). In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.\(L_p\)-\(L_q\) Fourier multipliers on locally compact quantum groupshttps://zbmath.org/1534.430032024-06-14T15:52:26.737412Z"Zhang, Haonan"https://zbmath.org/authors/?q=ai:zhang.haonanAmong numerous interesting results and examples the author proves the following theorems which generalize some famous results of Hörmander:
(1) Let \(1<p\leq 2\leq q <\infty\). Let \(\mathbb{G}\) be a locally compact quantum group with its dual \(\widehat{\mathbb{G}}\) such that the left Haar weight \(\varphi\) and the dual left Haar weight \(\widehat{\varphi}\) are tracial. Then for each \(x\in L_{r,\infty} (\mathbb{G},\varphi)\) with \(1/r=1/p-1/q\), \(m_x\) is an \(L_p\)-\(L_q\) Fourier multiplier such that \(\|m_x\|\lesssim_{p,q} \|x\|\).
(2) Let \(1<p\leq 2\). Let \(\mathbb{G}\) be a locally compact quantum group with its dual \(\widehat{\mathbb{G}}\) such that \(\varphi\) and \(\widehat{\varphi}\) are tracial. Then
\[
\|a\mathcal{F}(x)\|\lesssim \|a\|\cdot \|x\|,
\]
for all \(a \in L_{s,\infty} (\widehat{\mathbb{G}},\widehat{\varphi})\) and \(x\in L_{p} (\mathbb{G},\varphi)\), where \(1/s=2/p-1\).
Reviewer: Saak S. Gabriyelyan (Beer-Sheva)Apollonian packings and Kac-Moody root systemshttps://zbmath.org/1534.520192024-06-14T15:52:26.737412Z"Whitehead, Ian"https://zbmath.org/authors/?q=ai:whitehead.ianSummary: We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system \(\Phi \). We introduce the generating function \(Z(\mathbf{s})\) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for \(\Phi \). By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of \(\Phi \), with automorphic Weyl denominators, we express \(Z(\mathbf{s})\) in terms of Jacobi theta functions and the Siegel modular form \(\Delta_5\). We also show that the domain of convergence of \(Z(\mathbf{s})\) is the Tits cone of \(\Phi \), and discover that this domain inherits the intricate geometric structure of Apollonian packings.