Algebra & Number Theory最新文献

{"title":"On Kato and Kuzumaki’s properties for the Milnor K2 of function fields of p-adic curves","authors":"Diego Izquierdo, Giancarlo Lucchini Arteche","doi":"10.2140/ant.2024.18.815","DOIUrl":"https://doi.org/10.2140/ant.2024.18.815","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> be the function field of a curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> over a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. We prove that, for each <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>,</mo><mi>d</mi>\u0000<mo>≥</mo> <mn>1</mn></math> and for each hypersurface <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>ℙ</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>≤</mo>\u0000<mi>n</mi></math>, the second Milnor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>-theory group of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> is spanned by the images of the norms coming from finite extensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> over which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> has a rational point. When the curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> has a point in the maximal unramified extension of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>, we generalize this result to hypersurfaces <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>ℙ</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≤</mo>\u0000<mi>n</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"135 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976857","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 categorical Künneth formula for constructible Weil sheaves","authors":"Tamir Hemo, Timo Richarz, Jakob Scholbach","doi":"10.2140/ant.2024.18.499","DOIUrl":"https://doi.org/10.2140/ant.2024.18.499","url":null,"abstract":"<p>We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>&gt;</mo> <mn>0</mn></math> for various coefficients, including finite discrete rings, algebraic field extensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi>\u0000<mo>⊃</mo> <msub><mrow><mi>ℚ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi><mo>≠</mo><mi>p</mi></math>, and their rings of integers <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>E</mi></mrow></msub></math>. We also consider a variant for ind-constructible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898776","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":"Quotients of admissible formal schemes and adic spaces by finite groups","authors":"Bogdan Zavyalov","doi":"10.2140/ant.2024.18.409","DOIUrl":"https://doi.org/10.2140/ant.2024.18.409","url":null,"abstract":"<p>We give a self-contained treatment of finite group quotients of admissible (formal) schemes and adic spaces that are locally topologically finite type over a locally strongly noetherian adic space. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"185 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898799","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":"Subconvexity bound for GL(3) × GL(2) L-functions : Hybrid level aspect","authors":"Sumit Kumar, Ritabrata Munshi, Saurabh Kumar Singh","doi":"10.2140/ant.2024.18.477","DOIUrl":"https://doi.org/10.2140/ant.2024.18.477","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math> be a <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> Hecke–Maass cusp form of prime level <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> be a <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> Hecke–Maass cuspform of prime level <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. We will prove a subconvex bound for the <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> Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-function <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo>,</mo><mi>F</mi>\u0000<mo>×</mo>\u0000<mi>f</mi><mo stretchy=\"false\">)</mo></math> in the level aspect for certain ranges of the parameters <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"185 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898784","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":"Generalized Igusa functions and ideal growth in nilpotent Lie rings","authors":"Angela Carnevale, Michael M. Schein, Christopher Voll","doi":"10.2140/ant.2024.18.537","DOIUrl":"https://doi.org/10.2140/ant.2024.18.537","url":null,"abstract":"<p>We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"232 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898768","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 Tamagawa numbers of CM tori","authors":"Pei-Xin Liang, Yasuhiro Oki, Hsin-Yi Yang, Chia-Fu Yu","doi":"10.2140/ant.2024.18.583","DOIUrl":"https://doi.org/10.2140/ant.2024.18.583","url":null,"abstract":"<p>We investigate the problem of computing Tamagawa numbers of CM tori. This problem arises naturally from the problem of counting polarized abelian varieties with commutative endomorphism algebras over finite fields, and polarized CM abelian varieties and components of unitary Shimura varieties in the works of Achter, Altug, Garcia and Gordon and of Guo, Sheu and Yu, respectively. We make a systematic study on Galois cohomology groups in a more general setting and compute the Tamagawa numbers of CM tori associated to various Galois CM fields. Furthermore, we show that every (positive or negative) power of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math> is the Tamagawa number of a CM tori, proving the analogous conjecture of Ono for CM tori. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"34 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898804","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":"p-groups, p-rank, and semistable reduction of coverings of curves","authors":"Yu Yang","doi":"10.2140/ant.2024.18.281","DOIUrl":"https://doi.org/10.2140/ant.2024.18.281","url":null,"abstract":"<p>We prove various explicit formulas concerning <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-coverings of pointed semistable curves over discrete valuation rings. In particular, we obtain a full generalization of Raynaud’s formula for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank of fibers over <span>nonmarked smooth </span>closed points in the case of <span>arbitrary </span>closed points. As an application, for abelian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-coverings, we give an affirmative answer to an open problem concerning boundedness of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-rank asked by Saïdi more than twenty years ago. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"5 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695844","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":"Decidability via the tilting correspondence","authors":"Konstantinos Kartas","doi":"10.2140/ant.2024.18.209","DOIUrl":"https://doi.org/10.2140/ant.2024.18.209","url":null,"abstract":"<p>We prove a relative decidability result for perfectoid fields. This applies to show that the fields <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>∕</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>∞</mi></mrow></msup>\u0000</mrow></msup><mo stretchy=\"false\">)</mo></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>∞</mi></mrow></msup></mrow></msub><mo stretchy=\"false\">)</mo></math> are (existentially) decidable relative to the perfect hull of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>ℚ</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msubsup></math> is (existentially) decidable relative to the perfect hull of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mover accent=\"false\"><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math>. We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"35 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695696","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":"Partial sums of typical multiplicative functions over short moving intervals","authors":"Mayank Pandey, Victor Y. Wang, Max Wenqiang Xu","doi":"10.2140/ant.2024.18.389","DOIUrl":"https://doi.org/10.2140/ant.2024.18.389","url":null,"abstract":"<p>We prove that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>x</mi>\u0000<mo>+</mo>\u0000<mi>H</mi><mo stretchy=\"false\">]</mo></math> matches the corresponding Gaussian moment, as long as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi>\u0000<mo>≪</mo>\u0000<mi>x</mi><mo>∕</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mn>2</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo>\u0000</mrow></msup></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi></math> tends to infinity with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math>. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>x</mi>\u0000<mo>+</mo>\u0000<mi>H</mi><mo stretchy=\"false\">]</mo></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi>\u0000<mo>≪</mo>\u0000<mi>X</mi><mo>∕</mo><msup><mrow><mo stretchy=\"false\">(</mo><mi>log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>X</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>W</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></mrow></msup></math> tending to infinity with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math> is uniformly chosen from <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo><mo>,</mo><mi>X</mi><mo stretchy=\"false\">}</mo></math>, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>W</mi><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></math> tends to infinity with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> arbitrarily slowly. This makes some initial progress on a recent question of Harper. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695819","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 deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers","authors":"Chi-Yu Cheng","doi":"10.2140/ant.2024.18.319","DOIUrl":"https://doi.org/10.2140/ant.2024.18.319","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> be a representation of an acyclic quiver <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Q</mi></math> over an infinite field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. We establish a deterministic algorithm for computing the Harder–Narasimhan filtration of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>. The algorithm is polynomial in the dimensions of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>, the weights that induce the Harder–Narasimhan filtration of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>, and the number of paths in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Q</mi></math>. As a direct application, we also show that when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math> is algebraically closed and when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> is unstable, the same algorithm produces Kempf’s maximally destabilizing one parameter subgroups for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695805","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}