Algebra & Number Theory最新文献

{"title":"The diagonal coinvariant ring of a complex reflection group","authors":"Stephen Griffeth","doi":"10.2140/ant.2023.17.2033","DOIUrl":"https://doi.org/10.2140/ant.2023.17.2033","url":null,"abstract":"<p>For an irreducible complex reflection group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math> of rank <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> containing <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> reflections, we put <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>g</mi>\u0000<mo>=</mo> <mn>2</mn><mi>N</mi><mo>∕</mo><mi>n</mi></math> and construct a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math>-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the diagonal coinvariant ring of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math>. We propose that this representation of the Cherednik algebra is the single largest representation bearing this relationship to the diagonal coinvariant ring, and that further corrections to this estimate of the dimension of the diagonal coinvariant ring by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>g</mi>\u0000<mo>+</mo> <mn>1</mn><mo stretchy=\"false\">)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> should be orders of magnitude smaller. A crucial ingredient in the construction is the existence of a dot action of a certain product of symmetric groups (the Namikawa–Weyl group) acting on the parameter space of the rational Cherednik algebra and leaving invariant both the finite Hecke algebra and the spherical subalgebra; this fact is a consequence of ideas of Berest and Chalykh on the relationship between the Cherednik algebra and quasiinvariants. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"54 48","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514515","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":"Separation of periods of quartic surfaces","authors":"Pierre Lairez, Emre Can Sertöz","doi":"10.2140/ant.2023.17.1753","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1753","url":null,"abstract":"<p>We give a computable lower bound for the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether–Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 14","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493569","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":"Hybrid subconvexity bounds for twists of GL(3) × GL(2) L-functions","authors":"Bingrong Huang, Zhao Xu","doi":"10.2140/ant.2023.17.1715","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1715","url":null,"abstract":"<p>We prove hybrid subconvexity bounds 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>3</mn><mo stretchy=\"false\">)</mo>\u0000<mo>×</mo><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>\u0000<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions twisted by a primitive Dirichlet character modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> (prime) in the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>- and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>t</mi></math>-aspects. We also improve hybrid subconvexity bounds for twists of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math>\u0000<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions in the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>- and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>t</mi></math>-aspects. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 16","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493570","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":"Special cycles on the basic locus of unitary Shimura varieties at ramified primes","authors":"Yousheng Shi","doi":"10.2140/ant.2023.17.1681","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1681","url":null,"abstract":"<p>We study special cycles on the basic locus of certain unitary Shimura varieties over the ramified primes and their local analogs on the corresponding Rapoport–Zink spaces. We study the support and compute the dimension of these cycles. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 15","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493571","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 fake linear cycles inside Fermat varieties","authors":"Jorge Duque Franco, Roberto Villaflor Loyola","doi":"10.2140/ant.2023.17.1847","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1847","url":null,"abstract":"<p>We introduce a new class of Hodge cycles with nonreduced associated Hodge loci; we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></math>, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension, contrary to a conjecture of Movasati. They provide examples of algebraic cycles not generated by their periods in the sense of Movasati and Sertöz (2021). To study them we compute their Galois action in cohomology and their second-order invariant of the IVHS. We conclude that for any degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo> <mn>2</mn>\u0000<mo>+</mo> <mfrac><mrow><mn>6</mn></mrow>\u0000<mrow><mi>n</mi></mrow></mfrac></math>, the minimal codimension component of the Hodge locus passing through the Fermat variety is the one parametrizing hypersurfaces containing linear subvarieties of dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mrow><mi>n</mi></mrow>\u0000<mrow><mn>2</mn></mrow></mfrac> </math>, extending results of Green, Voisin, Otwinowska and the Villaflor Loyola. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 26","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493244","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":"Global dimension of real-exponent polynomial rings","authors":"Nathan Geist, Ezra Miller","doi":"10.2140/ant.2023.17.1779","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1779","url":null,"abstract":"<p>The ring <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math> of real-exponent polynomials in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> variables over any field has global dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>+</mo> <mn>1</mn></math> and flat dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math>. In particular, the residue field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits-->\u0000<mo>=</mo>\u0000<mi>R</mi><mo>∕</mo><mi mathvariant=\"fraktur\">𝔪</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math> modulo its maximal graded ideal <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"fraktur\">𝔪</mi></math> has flat dimension <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> via a Koszul-like resolution. Projective and flat resolutions of all <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math>-modules are constructed from this resolution of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></math>. The same results hold when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math> is replaced by the monoid algebra for the positive cone of any subgroup of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> satisfying a mild density condition. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 24","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493567","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":"Differences between perfect powers: prime power gaps","authors":"Michael A. Bennett, Samir Siksek","doi":"10.2140/ant.2023.17.1789","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1789","url":null,"abstract":"<p>We develop machinery to explicitly determine, in many instances, when the difference <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><mi>n</mi></mrow></msup></math> is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit equations, and the primitive divisor theorem of Bilu, Hanrot and Voutier) and classical algebraic number theory, with results derived from the modularity of Galois representations attached to Frey–Hellegoaurch elliptic curves. By way of example, we completely solve the equation </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo> <msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup>\u0000<mo>=</mo> <msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo>\u0000</math>\u0000</div>\u0000<p> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn>\u0000<mo>≤</mo>\u0000<mi>q</mi>\u0000<mo>&lt;</mo> <mn>1</mn><mn>0</mn><mn>0</mn></math> is prime, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>α</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> are integers with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>3</mn></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> gcd</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <mn>1</mn></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 25","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493566","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":"Unipotent ℓ-blocks for simply connected p-adic groups","authors":"Thomas Lanard","doi":"10.2140/ant.2023.17.1533","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1533","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math> be a nonarchimedean local field and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>-points of a connected simply connected reductive group over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>. We study the unipotent <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-blocks of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>, for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi><mo>≠</mo><mi>p</mi></math>. To that end, we introduce the notion of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-Harish-Chandra theory. The <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-blocks are then constructed using these <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>-series, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> the order of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math> modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>, and consistent systems of idempotents on the Bruhat–Tits building of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>. We also describe the stable <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi></math>-block decomposition of the depth zero category of an unramified classical group. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 4","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493240","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":"Isotriviality, integral points, and primitive primes in orbits in characteristic p","authors":"Alexander Carney, Wade Hindes, Thomas J. Tucker","doi":"10.2140/ant.2023.17.1573","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1573","url":null,"abstract":"<p>We prove a characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. In characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>, the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define and compare three different definitions of isotriviality for maps, sets, and curves. Using results of Favre and Rivera-Letelier on the structure of Julia sets, we prove that if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>φ</mi></math> is a nonisotrivial rational function and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>β</mi></math> is not exceptional for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>φ</mi></math>, then <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>φ</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>β</mi><mo stretchy=\"false\">)</mo></math> is a nonisotrivial set for all sufficiently large <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math>; we then apply diophantine results of Voloch and Wang that apply for all nonisotrivial sets. When <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>φ</mi></math> is a polynomial, we use the nonisotriviality of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>φ</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>β</mi><mo stretchy=\"false\">)</mo></math> for large <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> along with a partial converse to a result of Grothendieck in descent theory to deduce the nonisotriviality of the curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>y</mi></mrow><mrow><mi>ℓ</mi></mrow></msup>\u0000<mo>=</mo> <msup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo>\u0000<mo>−</mo>\u0000<mi>β</mi></math> for large <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> and small primes <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi><mo>≠</mo><mi>p</mi></math> whenever <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>β</mi></math> is not postcritical; this enables us to prove stronger results on Zsigmondy sets. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 5","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493239","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 structure of Frobenius kernels for automorphism group schemes","authors":"Stefan Schröer, Nikolaos Tziolas","doi":"10.2140/ant.2023.17.1637","DOIUrl":"https://doi.org/10.2140/ant.2023.17.1637","url":null,"abstract":"<p>We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristic. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with its twisted forms. The result actually holds true for arbitrary proper integral schemes under the assumption that the Frobenius kernel has large isotropy group at the generic point. This property is measured by a new numerical invariant called the foliation rank. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 7","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493237","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}