{"title":"Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic","authors":"Thibault Alexandre","doi":"10.2140/ant.2025.19.143","DOIUrl":"https://doi.org/10.2140/ant.2025.19.143","url":null,"abstract":"<p>We prove vanishing results for the coherent cohomology of the good reduction modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math> of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>λ</mi></math> near the walls of the antidominant Weyl chamber, there is an integer <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>e</mi>\u0000<mo>≥</mo> <mn>0</mn></math> such that the cohomology is concentrated in degrees <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><mn>0</mn><mo>,</mo><mi>e</mi><mo stretchy=\"false\">]</mo></math>. The accessible weights with our method are not necessarily regular and not necessarily <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-small. Since our method is technical, we also provide an algorithm written in SageMath that computes explicitly the vanishing results. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776368","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":"Picard rank jumps for K3 surfaces with bad reduction","authors":"Salim Tayou","doi":"10.2140/ant.2025.19.77","DOIUrl":"https://doi.org/10.2140/ant.2025.19.77","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> be a K3 surface over a number field. We prove that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar, Shankar and Tang to the case where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> has bad reduction. We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over <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></math> which extends previous work of Maulik, Shankar and Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"215 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776369","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 modification of the linear sieve, and the count of twin primes","authors":"Jared Duker Lichtman","doi":"10.2140/ant.2025.19.1","DOIUrl":"https://doi.org/10.2140/ant.2025.19.1","url":null,"abstract":"<p>We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math> in arithmetic progressions to moduli up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>1</mn><mn>0</mn><mo>∕</mo><mn>1</mn><mn>7</mn></mrow></msup></math>. This surpasses the level of distribution <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn><mo>∕</mo><mn>7</mn></mrow></msup></math> with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>x</mi></mrow><mrow><mn>7</mn><mo>∕</mo><mn>1</mn><mn>2</mn></mrow></msup></math> by Maynard. As an application, we obtain a new upper bound on the count of twin primes. Our method simplifies the 2004 argument of Wu, and gives the largest percentage improvement since the 1986 bound of Bombieri, Friedlander, and Iwaniec. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"28 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776367","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":"Ranks of abelian varieties in cyclotomic twist families","authors":"Ari Shnidman, Ariel Weiss","doi":"10.2140/ant.2025.19.39","DOIUrl":"https://doi.org/10.2140/ant.2025.19.39","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> be an abelian variety over a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math>, and suppose that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℤ</mi><mo stretchy=\"false\">[</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">]</mo></math> embeds in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> End</mi><mo> <!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mover accent=\"true\"><mrow><mi>F</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow></msub><mi>A</mi></math>, for some root of unity <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> of order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>=</mo> <msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup></math>. Assuming that the Galois action on the finite group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi><mo stretchy=\"false\">[</mo><mn>1</mn>\u0000<mo>−</mo> <msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">]</mo></math> is sufficiently reducible, we bound the average rank of the Mordell–Weil groups <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>F</mi><mo stretchy=\"false\">)</mo></math>, as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub></math> varies through the family of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>-twists of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math>. Combining this result with the recently proved uniform Mordell–Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup>\u0000<mo>=</mo>\u0000<mi>f</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math>, as well as in twist families of theta divisors of cyclic trigonal curves <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup>\u0000<mo>=</mo>\u0000<mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math>. Our main technical result is the determination of the average size of a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math>-isogeny Selmer group in a family of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mr","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"262 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776372","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":"Curves with few bad primes over cyclotomic ℤℓ-extensions","authors":"Samir Siksek, Robin Visser","doi":"10.2140/ant.2025.19.113","DOIUrl":"https://doi.org/10.2140/ant.2025.19.113","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> be a number field, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> a finite set of nonarchimedean places of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>, and write <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mo>×</mo></mrow></msup></math> for the group of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>. A famous theorem of Siegel asserts that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi>\u0000<mo>+</mo>\u0000<mi>δ</mi>\u0000<mo>=</mo> <mn>1</mn></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>δ</mi>\u0000<mo>∈</mo><msup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mo>×</mo></mrow></msup></math>, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> with good reduction outside <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>. Now instead of a number field, let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi>\u0000<mo>=</mo> <msub><mrow><mi>ℚ</mi></mrow><mrow><mi>∞</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math> which denotes the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>-cyclotomic extension of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math>. We show that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit equation <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi>\u0000<mo>+</mo>\u0000<mi>δ</mi>\u0000<mo>=</mo> <mn>1</mn></math>, with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>𝜀</mi></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>δ</mi>\u0000<mo>∈</mo><msup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mo>×</mo></mrow></msup></math>, has infinitely many solutions for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi>\u0000<mo>∈</mo><mo stretchy=\"false\">{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo stretchy=\"false\">}</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math> consists only of the totally ramified prime above <math display=\"inline\" xmlns=\"http:","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"47 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776370","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":"Super-Hölder vectors and the field of norms","authors":"Laurent Berger, Sandra Rozensztajn","doi":"10.2140/ant.2025.19.195","DOIUrl":"https://doi.org/10.2140/ant.2025.19.195","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math> be a field of characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. In a previous paper of ours, we defined and studied super-Hölder vectors in certain <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-linear representations of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>ℤ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>. In the present paper, we define and study super-Hölder vectors in certain <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-linear representations of a general <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic Lie group. We then consider certain <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic Lie extensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>∕</mo><mi>K</mi></math> of 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>, and compute the super-Hölder vectors in the tilt of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub></math>. We show that these super-Hölder vectors are the perfection of the field of norms of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>∕</mo><mi>K</mi></math>. By specializing to the case of a Lubin–Tate extension, we are able to recover <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi><mo stretchy=\"false\">(</mo><mo stretchy=\"false\">(</mo><mi>Y</mi>\u0000<mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math> inside the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Y</mi> </math>-adic completion of its perfection, seen as a valued <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi></math>-vector space endowed with the action of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math> given by the endomorphisms of the corresponding Lubin–Tate group. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776377","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":"Moduli of linear slices of high degree smooth hypersurfaces","authors":"Anand Patel, Eric Riedl, Dennis Tseng","doi":"10.2140/ant.2024.18.2133","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2133","url":null,"abstract":"<p>We study the variation of linear sections of hypersurfaces in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> hypersurface in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> varies maximally for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo>\u0000<mi>n</mi>\u0000<mo>+</mo> <mn>3</mn></math>. In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-planes in projective space and to free rational curves on arbitrary varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"225 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487002","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":"Separating G2-invariants of several octonions","authors":"Artem Lopatin, Alexandr N. Zubkov","doi":"10.2140/ant.2024.18.2157","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2157","url":null,"abstract":"<p>We describe separating <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-invariants of several copies of the algebra of octonions in case of a field of odd characteristic. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486628","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":"Matrix Kloosterman sums","authors":"Márton Erdélyi, Árpád Tóth","doi":"10.2140/ant.2024.18.2247","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2247","url":null,"abstract":"<p>We study a family of exponential sums that arises in the study of expanding horospheres on <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></math>. We prove an explicit version of general purity and find optimal bounds for these sums. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"210 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487037","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":"Scattering diagrams for generalized cluster algebras","authors":"Lang Mou","doi":"10.2140/ant.2024.18.2179","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2179","url":null,"abstract":"<p>We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"66 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486664","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
