Christopher H. Chiu, Alessandro Danelon, Jan Draisma, Rob H. Eggermont, Azhar Farooq
{"title":"Sym-Noetherianity for powers of GL-varieties","authors":"Christopher H. Chiu, Alessandro Danelon, Jan Draisma, Rob H. Eggermont, Azhar Farooq","doi":"10.2140/ant.2025.19.2091","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2091","url":null,"abstract":"<p>Much recent literature concerns finiteness properties of infinite-dimensional algebraic varieties equipped with an action of the infinite symmetric group, or of the infinite general linear group. In this paper, we study a common generalisation in which the product of both groups acts on infinite-dimensional spaces, and we show that these spaces are topologically Noetherian with respect to this action. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"35 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059681","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 boundedness of canonical models","authors":"Junpeng Jiao","doi":"10.2140/ant.2025.19.2119","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2119","url":null,"abstract":"<p>It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We confirm this conjecture when a general fiber of the corresponding Iitaka fibration is in a fixed bounded family of polarized log Calabi–Yau pairs. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"29 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059683","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":"Points of bounded height on certain subvarieties of toric varieties","authors":"Marta Pieropan, Damaris Schindler","doi":"10.2140/ant.2025.19.2281","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2281","url":null,"abstract":"<p>We combine the split torsor method and the hyperbola method for toric varieties to count rational points and Campana points of bounded height on certain subvarieties of toric varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"34 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059687","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":"Geometry of PCF parameters in spaces of quadratic polynomials","authors":"Laura DeMarco, Niki Myrto Mavraki","doi":"10.2140/ant.2025.19.2163","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2163","url":null,"abstract":"<p>We study algebraic relations among postcritically finite (PCF) parameters in the family <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo>\u0000<mo>=</mo> <msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>+</mo>\u0000<mi>c</mi></math>. It is known that an algebraic curve in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> contains infinitely many PCF pairs <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo></math> if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of arbitrary dimension in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> for any <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>2</mn></math>. Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> and the number of PCF parameters in real algebraic curves in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℂ</mi></math>, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>=</mo> <mn>1</mn></math>, we prove that there are only finitely many nonspecial lines in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℂ</mi>\u0000<mo>=</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> containing more than two PCF parameters. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059684","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":"An asymptotic orthogonality relation for GL(n, ℝ)","authors":"Dorian Goldfeld, Eric Stade, Michael Woodbury","doi":"10.2140/ant.2025.19.2185","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2185","url":null,"abstract":"<p>Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Asymptotic orthogonality relations for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≤</mo> <mn>3</mn></math>, and applications to number theory, have been considered by various researchers over the last 45 years. Recently, the authors of the present work have derived an explicit asymptotic orthogonality relation, with a power savings error term, 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>4</mn><mo>,</mo>\u0000<mi>ℝ</mi><mo stretchy=\"false\">)</mo></math>. Here we extend those results to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>n</mi><mo>,</mo>\u0000<mi>ℝ</mi><mo stretchy=\"false\">)</mo></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>2</mn></math>. </p><p> For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≤</mo> <mn>5</mn></math>, our results are contingent on the Ramanujan conjecture at the infinite place, but otherwise are unconditional. In particular, the case <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>=</mo> <mn>5</mn></math> represents a new result. The key new ingredient for the proof of the case <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>=</mo> <mn>5</mn></math> is the theorem of Kim and Shahidi that functorial products of cusp forms on <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>\u0000<mo>×</mo><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math> are automorphic on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>6</mn><mo stretchy=\"false\">)</mo></math>. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>></mo> <mn>5</mn></math> (assuming again the Raman","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059685","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 equivalence between the effective adjunction conjectures of Prokhorov–Shokurov and of Li","authors":"Jingjun Han, Jihao Liu, Qingyuan Xue","doi":"10.2140/ant.2025.19.2261","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2261","url":null,"abstract":"<p>Prokhorov and Shokurov introduced the effective adjunction conjecture, also known as the effective basepoint-freeness conjecture, which asserts that the moduli component of an lc-trivial fibration is effectively basepoint-free. Li proposed a variation of this conjecture, known as the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Γ</mi></math>-effective adjunction conjecture, and demonstrated that a weaker version of his conjecture follows from the original Prokhorov–Shokurov conjecture. </p><p> In this paper, we prove the equivalence between Prokhorov–Shokurov’s and Li’s effective adjunction conjectures. The key to our proof is establishing uniform rational polytopes for canonical bundle formulas. This relies on recent advancements in the minimal model program theory of algebraically integrable foliations, primarily developed by Ambro–Cascini–Shokurov–Spicer and Chen–Han–Liu–Xie. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"38 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145059686","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":"Paucity of rational points on fibrations with multiple fibres","authors":"Tim Browning, Julian Lyczak, Arne Smeets","doi":"10.2140/ant.2025.19.2049","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2049","url":null,"abstract":"<p>Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"24 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002871","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 via the double Bruhat graph, II : Iwahori–Hecke algebra","authors":"Felix Schremmer","doi":"10.2140/ant.2025.19.2015","DOIUrl":"https://doi.org/10.2140/ant.2025.19.2015","url":null,"abstract":"<p>We introduce a new language to describe the geometry of affine Deligne–Lusztig varieties in affine flag varieties. This second part of a two-paper series uses this new language, i.e., the double Bruhat graph, to describe certain structure constants of the Iwahori–Hecke algebra. As an application, we describe nonemptiness and dimension of affine Deligne–Lusztig varieties for most elements of the affine Weyl group and arbitrary <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>σ</mi></math>-conjugacy classes. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"111 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002870","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":"Smooth numbers are orthogonal to nilsequences","authors":"Lilian Matthiesen, Mengdi Wang","doi":"10.2140/ant.2025.19.1881","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1881","url":null,"abstract":"<p>The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><mi>y</mi><mo stretchy=\"false\">]</mo></math>-smooth if all of its prime factors belong to the interval <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><mi>y</mi><mo stretchy=\"false\">]</mo></math>. We identify suitable weights <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>g</mi></mrow><mrow><mo stretchy=\"false\">[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><mi>y</mi><mo stretchy=\"false\">]</mo></mrow></msub><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math> for the characteristic function of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><mi>y</mi><mo stretchy=\"false\">]</mo></math>-smooth numbers that allow us to establish strong asymptotic results on their distribution in short arithmetic progressions. Building on these equidistribution properties, we show that (a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi></math>-tricked version of) the function <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>g</mi></mrow><mrow><mo stretchy=\"false\">[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup><mo>,</mo><mi>y</mi><mo stretchy=\"false\">]</mo></mrow></msub><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo>\u0000<mo>−</mo> <mn>1</mn></math> is orthogonal to nilsequences. Our results apply in the almost optimal range <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mo stretchy=\"false\">(</mo><mi>log</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>N</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>K</mi></mrow></msup>\u0000<mo><</mo>\u0000<mi>y</mi>\u0000<mo>≤</mo>\u0000<mi>N</mi></math> of the smoothness parameter <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>y</mi></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi>\u0000<mo>≥</mo> <mn>2</mn></math> is sufficiently large, and to any <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>y</mi></mrow><mrow><mi>′</mi></mrow></msup>\u0000<mo><</mo><mi> min</mi><mo> <!--FUNCTION APPLICATION--> </mo><mo stretchy=\"false\">(</mo><msqrt><mrow><mi>y</mi></mrow></msqrt><mo>,</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>log</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>N</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>c</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>. </p><p> As a first application, we e","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002872","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 via the double Bruhat graph, I : Semi-infinite orbits","authors":"Felix Schremmer","doi":"10.2140/ant.2025.19.1973","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1973","url":null,"abstract":"<p>We introduce a new language to describe the geometry of affine Deligne–Lusztig varieties in affine flag varieties. This first part of a two-paper series develops the definition and fundamental properties of the double Bruhat graph by studying semi-infinite orbits. This double Bruhat graph was originally introduced by Naito and Watanabe to study periodic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>R</mi></math>-polynomials. We use it to describe the geometry of many affine Deligne–Lusztig varieties, overcoming a previously ubiquitous regularity condition. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145002869","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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