Algebra & Number Theory最新文献

{"title":"Maximal subgroups of exceptional groups and Quillen’s dimension","authors":"Kevin I. Piterman","doi":"10.2140/ant.2024.18.1375","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1375","url":null,"abstract":"<p>Given a finite group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> and a prime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>, let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒜</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math> be the poset of nontrivial elementary abelian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-subgroups of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>. The group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> satisfies the Quillen dimension property at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒜</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math> has nonzero homology in the maximal possible degree, which is the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> minus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>. For example, D. Quillen showed that solvable groups with trivial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-core satisfy this property, and later, M. Aschbacher and S. D. Smith provided a list of all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-extensions of simple groups that may fail this property if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> is odd. In particular, a group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> with this property satisfies Quillen’s conjecture: <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> has trivial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-core and the poset <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒜</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math> is not contractible. </p><p> In this article, we focus on the prime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>=</mo> <mn>2</mn></math> and prove that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>-extensions of finite simple groups of exceptional Lie type in odd characteristic satisfy the Quillen dimension property, wit","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Serre weights for three-dimensional wildly ramified Galois representations","authors":"Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra","doi":"10.2140/ant.2024.18.1221","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1221","url":null,"abstract":"<p>We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> Galois representations under a genericity condition when the field is unramified at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. This removes the assumption made previously that the representation be tamely ramified at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. We also prove a version of Breuil’s lattice conjecture and a mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> multiplicity one result for the cohomology of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>U</mi><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math>-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023). </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polyhedral and tropical geometry of flag positroids","authors":"Jonathan Boretsky, Christopher Eur, Lauren Williams","doi":"10.2140/ant.2024.18.1333","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1333","url":null,"abstract":"&lt;p&gt;A &lt;span&gt;flag positroid &lt;/span&gt;of ranks &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;&lt;&lt;/mo&gt;\u0000&lt;mo&gt;⋯&lt;/mo&gt;\u0000&lt;mo&gt;&lt;&lt;/mo&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; on &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt; is a flag matroid that can be realized by a real &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;×&lt;/mo&gt;\u0000&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt; matrix &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt; such that the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;×&lt;/mo&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; minors of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt; involving rows &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;…&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; are nonnegative for all &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mn&gt;1&lt;/mn&gt;\u0000&lt;mo&gt;≤&lt;/mo&gt;\u0000&lt;mi&gt;i&lt;/mi&gt;\u0000&lt;mo&gt;≤&lt;/mo&gt;\u0000&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;\u0000&lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;…&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt; TrFl&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt; equals the nonnegative flag Dressian &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt; FlDr&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;, and that the points &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;μ&lt;/mi&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;…&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Refined height pairing","authors":"Bruno Kahn","doi":"10.2140/ant.2024.18.1039","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1039","url":null,"abstract":"<p>For a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-dimensional regular proper variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> over the function field of a smooth variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> over a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> and for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>i</mi>\u0000<mo>≥</mo> <mn>0</mn></math>, we define a subgroup <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi> CH</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi> CH</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></math> and construct a “refined height pairing” </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<msup><mrow><mi>CH</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup>\u0000<mo>×</mo><msup><mrow><mi> CH</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup>\u0000<mo>→</mo><msup><mrow><mi> CH</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><mi>B</mi><mo stretchy=\"false\">)</mo>\u0000</math>\u0000</div>\u0000<p> in the category of abelian groups up to isogeny. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>i</mi>\u0000<mo>=</mo> <mn>1</mn><mo>,</mo><mi>d</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi> CH</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo></mrow></msup></math> is the group of cycles numerically equivalent to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn></math>. This pairing relates to pairings defined by P. Schneider and A. Beilinson if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> is a curve, to a refined height defined by ","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"70 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140817820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Balmer spectra and Drinfeld centers","authors":"Kent B. Vashaw","doi":"10.2140/ant.2024.18.1081","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1081","url":null,"abstract":"<p>The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>C</mi></mstyle></math> to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>C</mi></mstyle></math> extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson–Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"58 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140818076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the p-adic interpolation of unitary Friedberg–Jacquet periods","authors":"Andrew Graham","doi":"10.2140/ant.2024.18.1117","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1117","url":null,"abstract":"<p>We establish functoriality of higher Coleman theory for certain unitary Shimura varieties and use this to construct a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic analytic function interpolating unitary Friedberg–Jacquet periods. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"19 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140817727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Enumeration of conjugacy classes in affine groups","authors":"Jason Fulman, Robert M. Guralnick","doi":"10.2140/ant.2024.18.1189","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1189","url":null,"abstract":"<p>We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the noncoprime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mi>V</mi>\u0000<mo stretchy=\"false\">)</mo></math> problem of Brauer. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140817935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Locally analytic vector bundles on the Fargues–Fontaine curve","authors":"Gal Porat","doi":"10.2140/ant.2024.18.899","DOIUrl":"https://doi.org/10.2140/ant.2024.18.899","url":null,"abstract":"<p>We develop a version of Sen theory for equivariant vector bundles on the Fargues–Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>φ</mi><mo>,</mo><mi>Γ</mi><mo stretchy=\"false\">)</mo></math>-modules in the cyclotomic case then recovers the Cherbonnier–Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles. Using the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic monodromy theorem, we show that each locally analytic vector bundle <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">ℰ</mi></math> has a canonical differential equation for which the space of solutions has full rank. As a consequence, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">ℰ</mi></math> and its sheaf of solutions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Sol</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi mathvariant=\"bold-script\">ℰ</mi><mo stretchy=\"false\">)</mo></math> are in a natural correspondence, which gives a geometric interpretation of a result of Berger on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>φ</mi><mo>,</mo><mi>Γ</mi><mo stretchy=\"false\">)</mo></math>-modules. In particular, if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> is a de Rham Galois representation, its associated filtered <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>φ</mi><mo>,</mo><mi>N</mi><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mspace width=\"-0.17em\"></mspace><mi>K</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>-module is realized as the space of global solutions to the differential equation. A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate–Sen formalism, which is also of independent interest. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"48 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140556458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Theta correspondence and simple factors in global Arthur parameters","authors":"Chenyan Wu","doi":"10.2140/ant.2024.18.969","DOIUrl":"https://doi.org/10.2140/ant.2024.18.969","url":null,"abstract":"<p>By using results on poles of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions and theta correspondence, we give a bound on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>χ</mi><mo>,</mo><mi>b</mi><mo stretchy=\"false\">)</mo></math>-factors of the global Arthur parameter of a cuspidal automorphic representation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>π</mi></math> of a classical group or a metaplectic group where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>χ</mi></math> is a conjugate self-dual automorphic character and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> is an integer which is the dimension of an irreducible representation of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> SL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math>. We derive a more precise relation when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>π</mi></math> lies in a generic global <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math>-packet. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140556471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect","authors":"Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi","doi":"10.2140/ant.2024.18.993","DOIUrl":"https://doi.org/10.2140/ant.2024.18.993","url":null,"abstract":"<p>This paper is an extension of Kim et al. (2020a), and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur’s invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications, including the vertical Sato–Tate theorem and low-lying zeros for standard <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions of holomorphic Siegel cusp forms, are discussed. We also show that the “nongenuine forms”, which come from nontrivial endoscopic contributions by Langlands functoriality classified by Arthur, are negligible. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140556504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}