We develop an abstract framework for studying the strong form of Malle’s conjecture [J. Number Theory 92 (2002), pp. 315–329; Experiment. Math. 13 (2004), pp. 129–135] for nilpotent groups GG in their regular representation. This framework is then used to prove the strong form of Malle’s conjecture for any nilpotent group GG such that all elements of order pp are central, where pp is the smallest prime divisor of #G# G.
We also give an upper bound for any nilpotent group GG tight up to logarithmic factors, and tight up to a constant factor in case all elements of order pp pairwise commute. Finally, we give a new heuristical argument supporting Malle’s conjecture in the case of nilpotent groups in their regular representation.
{"title":"On Malle’s conjecture for nilpotent groups","authors":"P. Koymans, Carlo Pagano","doi":"10.1090/btran/140","DOIUrl":"https://doi.org/10.1090/btran/140","url":null,"abstract":"<p>We develop an abstract framework for studying the strong form of Malle’s conjecture [J. Number Theory 92 (2002), pp. 315–329; Experiment. Math. 13 (2004), pp. 129–135] for nilpotent groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in their regular representation. This framework is then used to prove the strong form of Malle’s conjecture for any nilpotent group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> such that all elements of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are central, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the smallest prime divisor of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"number-sign upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">#<!-- # --></mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\"># G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>\u0000\u0000<p>We also give an upper bound for any nilpotent group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> tight up to logarithmic factors, and tight up to a constant factor in case all elements of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> pairwise commute. Finally, we give a new heuristical argument supporting Malle’s conjecture in the case of nilpotent groups in their regular representation.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134021821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A polynomial in a single variable is uniquely determined by its derivatives of even order at 0 and 1. More precisely, such an univariate polynomial can be written and a finite sum of f(2n)(0)Λn(1−z)f^{(2n)}(0) Lambda _n(1-z) and f(2n)(1)Λn(z)f^{(2n)}(1) Lambda _n(z), (n≥0nge 0), where the
单变量多项式唯一地由它在0和1处的偶阶导数决定。更准确地说，这样的单变量多项式可以写成f (2n)(0) Λ n(1-z) f^{(2n)(0)}Lambda ＿n(1-z)和f (2n)(1) Λ n(z) f^{(2n)}(1) Lambda ＿n(z)，(n≥0 n ge 0)，其中Λ n(z) Lambda ＿n(z)是由条件(d d z) 2k Λ n(0) = 0和(D D z) 2k Λ n (1) = δ k, n, k≥0,n≥0。begin{equation*} left (frac {mathrm {d}}{mathrm {d}z}right )^{2k} Lambda _n(0)=0text { and } left (frac {mathrm {d}}{mathrm {d}z}right )^{2k} Lambda _n(1)=delta _{k,n},quad kge 0, ; nge 0. end{equation*}我们把这个理论推广到n n个变量，用n+1个+1个点e _ 0, e _ 1，…代替C mathbb中的两个点0 0,1 1，e＿n e＿0{, }e＿1{underline, {}}{underline{}}dots, e＿n {underlinein {
{"title":"Lidstone interpolation III. Several variables","authors":"M. Waldschmidt","doi":"10.1090/btran/135","DOIUrl":"https://doi.org/10.1090/btran/135","url":null,"abstract":"<p>A polynomial in a single variable is uniquely determined by its derivatives of even order at 0 and 1. More precisely, such an univariate polynomial can be written and a finite sum of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript left-parenthesis 2 n right-parenthesis Baseline left-parenthesis 0 right-parenthesis normal upper Lamda Subscript n Baseline left-parenthesis 1 minus z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>z</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">f^{(2n)}(0) Lambda _n(1-z)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript left-parenthesis 2 n right-parenthesis Baseline left-parenthesis 1 right-parenthesis normal upper Lamda Subscript n Baseline left-parenthesis z right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>z</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">f^{(2n)}(1) Lambda _n(z)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, (<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">nge 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>), where the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda Subscript n Baseline ","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"52 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134033691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an abelian number field of odd degree, we study the structure of its 2 2 -Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.
{"title":"On unit signatures and narrow class groups of odd degree abelian number fields","authors":"Benjamin Breen, Ila Varma, J. Voight","doi":"10.1090/btran/90","DOIUrl":"https://doi.org/10.1090/btran/90","url":null,"abstract":"For an abelian number field of odd degree, we study the structure of its \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"219 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122077753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We enumerate and classify all stationary logarithmic configurations of d+2d+2 points on the unit sphere in dd–dimensions. In particular, we show that the logarithmic energy attains its local minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality mm and nn. The global minimum occurs when m=nm=n if dd is even and m=n+1m=n+1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on S
{"title":"Log-optimal (𝑑+2)-configurations in 𝑑–dimensions","authors":"P. Dragnev, O. Musin","doi":"10.1090/btran/118","DOIUrl":"https://doi.org/10.1090/btran/118","url":null,"abstract":"<p>We enumerate and classify all stationary logarithmic configurations of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d plus 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">d+2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> points on the unit sphere in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>–dimensions. In particular, we show that the logarithmic energy attains its local minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\u0000 <mml:semantics>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The global minimum occurs when <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m equals n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">m=n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is even and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m equals n plus 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">m=n+1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Superscript d minus 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow ","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"167 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114152518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The space of normal measures on a measurable cardinal is naturally ordered by the Mitchell ordering. In the first part of this paper we show that the Mitchell ordering can be linear on a strong cardinal where the Generalised Continuum Hypothesis fails. In the second part we show that a supercompact cardinal at which the Generalised Continuum Hypothesis fails may carry a very large number of normal measures of Mitchell order zero.
{"title":"Normal measures on large cardinals","authors":"Arthur W. Apter, J. Cummings","doi":"10.1090/btran/132","DOIUrl":"https://doi.org/10.1090/btran/132","url":null,"abstract":"The space of normal measures on a measurable cardinal is naturally ordered by the Mitchell ordering. In the first part of this paper we show that the Mitchell ordering can be linear on a strong cardinal where the Generalised Continuum Hypothesis fails. In the second part we show that a supercompact cardinal at which the Generalised Continuum Hypothesis fails may carry a very large number of normal measures of Mitchell order zero.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131143403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let kk be any field and let GG be a connected reductive algebraic kk-group. Associated to GG is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] and Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] that is called the index of GG (a Dynkin diagram along with some additional combinatorial information). Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] showed that the kk-isogeny class of GG is uniquely determined by its index and the kk-isogeny class of
设k k是任意域，G G是一个连通的约化代数k k群。与G相关的G是20世纪60年代由Satake [Ann]首次研究的不变量。的数学。(2) 71(1960)， 77-110)和Tits [thsamorie des Groupes algbriques(布鲁塞尔，1962)，鲁汶大学图书馆;[代数群与不连续子群]，《数学学报》，2002。纯粹数学。，博尔德，科罗拉多州，1965)，美国。数学。Soc。， Providence, r.i.， 1966, pp. 33-62]，被称为G G的索引(带有一些附加组合信息的Dynkin图)。代数群与不连续子群。纯粹数学。，博尔德，科罗拉多州，1965)，美国。数学。Soc。(Providence, r.i.， 1966, pp. 33-62)证明了G G的k k等同性类是由它的指数和它的各向异性核G a G_a的k k等同性类唯一决定的。对于G G绝对简单的情况，G G索引的所有可能性都被Tits[代数群和不连续子群](Proc. Sympos)分类。纯粹数学。，博尔德，科罗拉多州，1965)，美国。数学。Soc。[j]，《普罗维登斯，罗德岛，1966》，第33-62页。设H H是G G中最大秩的连通约化k k子群。我们在G G中引入H H的G(k) G(k)共轭类的一个不变量，称为H∧G H 子集G的指标嵌入。它由H H索引和G G索引以及满足一定兼容条件的嵌入图组成。在G G的k k -子群集合上引入了一个称为指标共轭的等价关系，并观察到G G中H H的G(k) G(k) -共轭类是由G
{"title":"Maximal connected 𝑘-subgroups of maximal rank in connected reductive algebraic 𝑘-groups","authors":"Damian Sercombe","doi":"10.1090/btran/112","DOIUrl":"https://doi.org/10.1090/btran/112","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be any field and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a connected reductive algebraic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-group. Associated to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] and Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] that is called the index of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (a Dynkin diagram along with some additional combinatorial information). Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] showed that the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-isogeny class of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is uniquely determined by its index and the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-isogeny class of","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123652793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish two surprising types of Weyl’s laws for some compact RCD ( K , N ) operatorname {RCD}(K, N) /Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for RCD ( K , N ) operatorname {RCD}(K,N) spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the α alpha -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].
{"title":"Singular Weyl’s law with Ricci curvature bounded below","authors":"Xianche Dai, Shouhei Honda, Jiayin Pan, G. Wei","doi":"10.1090/btran/160","DOIUrl":"https://doi.org/10.1090/btran/160","url":null,"abstract":"We establish two surprising types of Weyl’s laws for some compact \u0000\u0000 \u0000 \u0000 RCD\u0000 \u0000 (\u0000 K\u0000 ,\u0000 N\u0000 )\u0000 \u0000 operatorname {RCD}(K, N)\u0000 \u0000\u0000/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for \u0000\u0000 \u0000 \u0000 RCD\u0000 \u0000 (\u0000 K\u0000 ,\u0000 N\u0000 )\u0000 \u0000 operatorname {RCD}(K,N)\u0000 \u0000\u0000 spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the \u0000\u0000 \u0000 α\u0000 alpha\u0000 \u0000\u0000-Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114320606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of certain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.
{"title":"Compact difference of composition operators on the Hardy spaces","authors":"B. Choe, Koeun Choi, H. Koo, Inyoung Park","doi":"10.1090/btran/126","DOIUrl":"https://doi.org/10.1090/btran/126","url":null,"abstract":"Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of certain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132521491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study boundaries for unital operator algebras. These are sets of irreducible ∗ * -representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of C ∗ mathrm {C}^* -algebras, our techniques allow us to provide a new proof of a recent characterization of those C ∗ mathrm {C}^* -algebras admitting only finite-dimensional irreducible representations.
{"title":"Minimal boundaries for operator algebras","authors":"Raphael Clouatre, I. Thompson","doi":"10.1090/btran/154","DOIUrl":"https://doi.org/10.1090/btran/154","url":null,"abstract":"We study boundaries for unital operator algebras. These are sets of irreducible \u0000\u0000 \u0000 ∗\u0000 *\u0000 \u0000\u0000-representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of \u0000\u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 ∗\u0000 \u0000 mathrm {C}^*\u0000 \u0000\u0000-algebras, our techniques allow us to provide a new proof of a recent characterization of those \u0000\u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 ∗\u0000 \u0000 mathrm {C}^*\u0000 \u0000\u0000-algebras admitting only finite-dimensional irreducible representations.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"854 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127604752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we show that the finite subalgebra AR(1)mathcal {A}^mathbb {R}(1), generated by Sq1mathrm {Sq}^1 and Sq2mathrm {Sq}^2, of the Rmathbb {R}-motivic Steenrod algebra ARmathcal {A}^mathbb {R} can be given 128 different
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