Algebra & Number Theory最新文献

{"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}

{"title":"Multiplicity structure of the arc space of a fat point","authors":"Rida Ait El Manssour, Gleb Pogudin","doi":"10.2140/ant.2024.18.947","DOIUrl":"https://doi.org/10.2140/ant.2024.18.947","url":null,"abstract":"<p>The equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup>\u0000<mo>=</mo> <mn>0</mn></math> defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi><mo stretchy=\"false\">[</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></mrow></msup><mo>,</mo><mi>…</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo><mo stretchy=\"false\">]</mo></math> by all differential consequences of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup>\u0000<mo>=</mo> <mn>0</mn></math>. This infinite-dimensional algebra admits a natural filtration by finite-dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>m</mi><mo>∕</mo><mo stretchy=\"false\">(</mo><mn>1</mn>\u0000<mo>−</mo>\u0000<mi>m</mi><mi>t</mi><mo stretchy=\"false\">)</mo></math>. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by the nonreduced version of the geometric motivic Poincaré series, multiplicities in differential algebra, and connections between arc spaces and the Rogers–Ramanujan identities. We also prove a recent conjecture put forth by Afsharijoo in the latter context. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140556548","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 ordinary Hecke orbit conjecture","authors":"Pol van Hoften","doi":"10.2140/ant.2024.18.847","DOIUrl":"https://doi.org/10.2140/ant.2024.18.847","url":null,"abstract":"<p>We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre–Tate coordinates of Chai as well as recent results of D’Addezio about the monodromy groups of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result for the formal completions of ordinary Hecke orbits. Along the way, we show that classical Serre–Tate coordinates can be described using unipotent formal groups, generalising a result of Howe. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"34 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140556579","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 Ozaki’s theorem realizing prescribed p-groups as p-class tower groups","authors":"Farshid Hajir, Christian Maire, Ravi Ramakrishna","doi":"10.2140/ant.2024.18.771","DOIUrl":"https://doi.org/10.2140/ant.2024.18.771","url":null,"abstract":"<p>We give a streamlined and effective proof of Ozaki’s theorem that any finite <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Γ</mi></math> is the Galois group of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-Hilbert class field tower of some number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></math>. Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub></math> with class number prime to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. We construct <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>∕</mo><msub><mrow><mi>k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub></math> by a sequence of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℤ</mi><mo>∕</mo><mi>p</mi></math>-extensions ramified only at finite tame primes and also give explicit bounds on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><mi>F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits-->\u0000<mo>:</mo><msub><mrow><mi> k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">]</mo></math> and the number of ramified primes of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>∕</mo><msub><mrow><mi>k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub></math> in terms of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>#</mi><mi>Γ</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"142 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976770","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":"Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields","authors":"Asbjørn Christian Nordentoft","doi":"10.2140/ant.2024.18.735","DOIUrl":"https://doi.org/10.2140/ant.2024.18.735","url":null,"abstract":"<p>We calculate certain “wide moments” of central values of Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi><mrow><mo fence=\"true\" mathsize=\"1.19em\">(</mo><mrow><mi>π</mi>\u0000<mo>⊗</mo><mi mathvariant=\"normal\">Ω</mi><mo>,</mo> <mfrac><mrow><mn>1</mn></mrow>\u0000<mrow><mn>2</mn></mrow></mfrac></mrow><mo fence=\"true\" mathsize=\"1.19em\">)</mo></mrow></math> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>π</mi></math> is a cuspidal automorphic representation of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>2</mn></mrow></msub></math> over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math> is a Hecke character (of conductor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions where the product of the twists is trivial. </p><p> The proof relies on relating the wide moments of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger’s formula. To achieve this, a classical version of Waldspurger’s formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals. In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976783","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":"Infinitesimal dilogarithm on curves over truncated polynomial rings","authors":"Sinan Ünver","doi":"10.2140/ant.2024.18.685","DOIUrl":"https://doi.org/10.2140/ant.2024.18.685","url":null,"abstract":"<p>We construct infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This generalizes Park’s work on the regulators of additive cycles. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976694","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":"Fundamental exact sequence for the pro-étale fundamental group","authors":"Marcin Lara","doi":"10.2140/ant.2024.18.631","DOIUrl":"https://doi.org/10.2140/ant.2024.18.631","url":null,"abstract":"<p>The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> ét</mtext><!--/mstyle--></mrow></msubsup></math> defined in SGA1 and the more general <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>SGA3</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow></msubsup></math>. It controls local systems in the pro-étale topology and leads to an interesting class of “geometric coverings” of schemes, generalizing finite étale coverings. </p><p> We prove exactness of the fundamental sequence for the pro-étale fundamental group of a geometrically connected scheme <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> of finite type over a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>, i.e., that the sequence </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mn>1</mn>\u0000<mo>→</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent=\"true\"><mrow>\u0000<mi>k</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow></msub><mo stretchy=\"false\">)</mo>\u0000<mo>→</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo>\u0000<mo>→</mo><msub><mrow><mi> Gal</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow>\u0000<mi>k</mi></mrow></msub>\u0000<mo>→</mo> <mn>1</mn>\u0000</math>\u0000</div>\u0000<p> is exact as abstract groups and the map <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent=\"true\"><mrow><mi>k</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow></msub><mo stretchy=\"false\">)</mo>\u0000<mo>→</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></math> is a topological embedding. </p><p> On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"30 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976765","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":"Supersolvable descent for rational points","authors":"Yonatan Harpaz, Olivier Wittenberg","doi":"10.2140/ant.2024.18.787","DOIUrl":"https://doi.org/10.2140/ant.2024.18.787","url":null,"abstract":"<p>We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer–Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the existence of supersolvable Galois extensions of number fields with prescribed norms, generalising work of Frei, Loughran and Newton. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976774","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}