Algebra & Number Theory最新文献

{"title":"Semistable models for some unitary Shimura varieties over ramified primes","authors":"Ioannis Zachos","doi":"10.2140/ant.2024.18.1715","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1715","url":null,"abstract":"<p>We consider Shimura varieties associated to a unitary group of signature <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>n</mi>\u0000<mo>−</mo> <mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>. We give regular <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic integral models for these varieties over odd primes <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> which ramify in the imaginary quadratic field with level subgroup at <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> given by the stabilizer of a selfdual lattice in the hermitian space. Our construction is given by an explicit resolution of a corresponding local model. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"197 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245250","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":"Application of a polynomial sieve: beyond separation of variables","authors":"Dante Bonolis, Lillian B. Pierce","doi":"10.2140/ant.2024.18.1515","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1515","url":null,"abstract":"&lt;p&gt;Let a polynomial &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;\u0000&lt;mi&gt;ℤ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&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;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt; be given. The square sieve can provide an upper bound for the number of integral &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; such that &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; for which &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; is solvable for &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;y&lt;/mi&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;\u0000&lt;mi&gt;ℤ&lt;/mi&gt;&lt;/math&gt;; then to a polynomial sieve, counting &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; for which &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;\u0000&lt;mo&gt;=&lt;/mo&gt;\u0000&lt;mi&gt;g&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is solvable, for a given polynomial &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;. Formally, a polynomial sieve lemma can encompass the more general problem of counting &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; for which &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stre","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236172","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":"Unramifiedness of weight 1 Hilbert Hecke algebras","authors":"Shaunak V. Deo, Mladen Dimitrov, Gabor Wiese","doi":"10.2140/ant.2024.18.1465","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1465","url":null,"abstract":"<p>We prove that the Galois pseudorepresentation valued in the mod <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math> cuspidal Hecke algebra for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> over a totally real number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, of parallel weight <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> and level prime to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>, is unramified at any place above <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. The same is true for the noncuspidal Hecke algebra at places above <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> whose ramification index is not divisible by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi><mo>−</mo><mn>1</mn></math>. A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> ramifies in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, of generalised <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Θ</mi></math>-operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"29 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236177","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":"Failure of the local-global principle for isotropy of quadratic forms over function fields","authors":"Asher Auel, V. Suresh","doi":"10.2140/ant.2024.18.1497","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1497","url":null,"abstract":"<p>We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math> variables over function fields of transcendence degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>2</mn></math> over an algebraically closed field of characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo>≠</mo><mn>2</mn></math>. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified cohomology of products of elliptic curves over discretely valued fields. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"8 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236173","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":"The strong maximal rank conjecture and moduli spaces of curves","authors":"Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang","doi":"10.2140/ant.2024.18.1403","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1403","url":null,"abstract":"<p>Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu–Farkas strong maximal rank conjecture, in genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>2</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>3</mn></math>. This constitutes a major step forward in Farkas’ program to prove that the moduli spaces of curves of genus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>2</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn><mn>3</mn></math> are of general type. Our techniques involve a combination of the Eisenbud–Harris theory of limit linear series, and the notion of linked linear series developed by Osserman. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"75 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236169","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":"Functorial embedded resolution via weighted blowings up","authors":"Dan Abramovich, Michael Temkin, Jarosław Włodarczyk","doi":"10.2140/ant.2024.18.1557","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1557","url":null,"abstract":"<p>We provide a simple procedure for resolving, in characteristic 0, singularities of a variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> embedded in a smooth variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Y</mi> </math> by repeatedly blowing up the worst singularities, in the sense of stack-theoretic weighted blowings up. No history, no exceptional divisors, and no logarithmic structures are necessary to carry this out; the steps are explicit geometric operations requiring no choices; and the resulting algorithm is efficient. </p><p> A similar result was discovered independently by McQuillan (2020). </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236174","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":"Exceptional characters and prime numbers in sparse sets","authors":"Jori Merikoski","doi":"10.2140/ant.2024.18.1305","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1305","url":null,"abstract":"<p>We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>b</mi></mrow><mrow><mn>8</mn></mrow></msup></math>. </p>","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":"141315704","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":"Combining Igusa’s conjectures on exponential sums and monodromy with semicontinuity of the minimal exponent","authors":"Raf Cluckers, Kien Huu Nguyen","doi":"10.2140/ant.2024.18.1275","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1275","url":null,"abstract":"<p>We combine two of Igusa’s conjectures with recent semicontinuity results by Mustaţă and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa’s conjecture on exponential sums, but also some new evidence like for all polynomials in up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math> variables. We show that, in turn, these bounds imply consequences for Igusa’s (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"59 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315546","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":"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":"&lt;p&gt;Given a finite group &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt; and a prime &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;, let &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;𝒜&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mspace width=\"-0.17em\"&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; be the poset of nontrivial elementary abelian &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;-subgroups of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;. The group &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt; satisfies the Quillen dimension property at &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt; if &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;𝒜&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mspace width=\"-0.17em\"&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; has nonzero homology in the maximal possible degree, which is the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;-rank of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt; minus &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;. For example, D. Quillen showed that solvable groups with trivial &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;-core satisfy this property, and later, M. Aschbacher and S. D. Smith provided a list of all &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;-extensions of simple groups that may fail this property if &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt; is odd. In particular, a group &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt; with this property satisfies Quillen’s conjecture: &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt; has trivial &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;-core and the poset &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"bold-script\"&gt;𝒜&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mspace width=\"-0.17em\"&gt;&lt;/mspace&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is not contractible. &lt;/p&gt;&lt;p&gt; In this article, we focus on the prime &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;p&lt;/mi&gt;\u0000&lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt; and prove that the &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;-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}