{"title":"The wavefront sets of unipotent supercuspidal representations","authors":"Dan Ciubotaru, Lucas Mason-Brown, Emile Okada","doi":"10.2140/ant.2024.18.1863","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1863","url":null,"abstract":"<p>We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic group is a singleton provided <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>></mo> <mn>3</mn><mo stretchy=\"false\">(</mo><mi>h</mi>\u0000<mo>−</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>h</mi></math> is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs to Lusztig’s category of unipotent representations, we give an explicit formula for the double and geometric wavefront sets. As a consequence, we show that the nilpotent part of the Deligne–Langlands–Lusztig parameter of a unipotent supercuspidal representation is precisely the image of its geometric wavefront set under Spaltenstein’s duality map. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"29 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384208","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":"Affine Deligne–Lusztig varieties with finite Coxeter parts","authors":"Xuhua He, Sian Nie, Qingchao Yu","doi":"10.2140/ant.2024.18.1681","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1681","url":null,"abstract":"<p>We study affine Deligne–Lusztig varieties <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub></math> when the finite part of the element <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> in the Iwahori–Weyl group is a partial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>σ</mi></math>-Coxeter element. We show that such <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> is a cordial element and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub><mo>≠</mo><mi>∅</mi></math> if and only if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>w</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub></math> has a simple geometric structure: the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>σ</mi></math>-centralizer of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> acts transitively on the set of irreducible components of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo stretchy=\"false\">(</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msub></math>; and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"double-struck\">𝔸</mi></mrow><mrow><mn>1</mn></mrow></msup></math> or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow></msub></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245248","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":"A unipotent realization of the chromatic quasisymmetric function","authors":"Lucas Gagnon","doi":"10.2140/ant.2024.18.1737","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1737","url":null,"abstract":"<p>We realize two families of combinatorial symmetric functions via the complex character theory of the finite general linear group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>: chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> UT</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>. The proof of these results also gives a general Hopf algebraic approach to computing the induction map. Additional results include a connection between the relevant <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> characters and Hessenberg varieties and a reinterpretation of known theorems and conjectures about the relevant symmetric functions in terms of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"112 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245253","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":"A bound for the exterior product of S-units","authors":"Shabnam Akhtari, Jeffrey D. Vaaler","doi":"10.2140/ant.2024.18.1589","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1589","url":null,"abstract":"<p>We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units contained in a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. This leads to a bound for the exterior product of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit group but not on the field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. Our inequality is related to a conjecture of F. Rodriguez Villegas. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"64 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245251","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":"Prime values of f(a,b2) and f(a,p2), f quadratic","authors":"Stanley Yao Xiao","doi":"10.2140/ant.2024.18.1619","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1619","url":null,"abstract":"<p>We prove an asymptotic formula for primes of the shape <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>a</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>b</mi></math> integers and of the shape <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> prime. Here <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> is a binary quadratic form with integer coefficients, irreducible over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math> and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>y</mi></mrow><mrow><mn>4</mn></mrow></msup></math> and of Heath-Brown and Li on 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>p</mi></mrow><mrow><mn>4</mn></mrow></msup></math>, as well as earlier work of the author with Lam and Schindler on primes of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo>,</mo><mi>p</mi><mo stretchy=\"false\">)</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> a positive definite form. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245246","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":"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":"<p>Let a polynomial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi>\u0000<mo>∈</mo>\u0000<mi>ℤ</mi><mo stretchy=\"false\">[</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo> <!--FUNCTION APPLICATION--></mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">]</mo></math> be given. The square sieve can provide an upper bound for the number of integral <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> such that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo></math> is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>r</mi></mrow></msup></math> is solvable for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>y</mi>\u0000<mo>∈</mo>\u0000<mi>ℤ</mi></math>; then to a polynomial sieve, counting <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo>\u0000<mi>g</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">)</mo></math> is solvable, for a given polynomial <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi></math>. Formally, a polynomial sieve lemma can encompass the more general problem of counting <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle>\u0000<mo>∈</mo> <msup><mrow><mo stretchy=\"false\">[</mo><mo>−</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo stretchy=\"false\">]</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo>,</mo><mstyle mathvariant=\"bold-italic\"><mi>x</mi></mstyle><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}
Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang
{"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}
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