We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size in arithmetic progressions to moduli up to . This surpasses the level of distribution with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to 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.
{"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}
<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> <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> <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> <mo>=</mo> <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> <mo>=</mo> <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
设A是数字域F上的一个阿贝尔变，并假设对于n阶的单位ζn = 3m的某个根，n [ζn]嵌入到End （F¯A）中。假设有限群A[1−ζn]上的伽罗瓦作用是充分可约的，我们对modell - weil群Ad(F)的平均秩进行了定界，当Ad在A的μ2n-扭转族中变化时，结合最近证明的一致modell - lang猜想，我们证明了双环三角曲线y3= F （x2）的扭转族中有理点个数的近似一致界，以及循环三角曲线y3= F (x)的θ因子扭转族中的有理点个数的近似一致界。我们的主要技术成果是确定μ2n-扭转家族中3-等基因Selmer基团的平均大小。
{"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}
Let be a field of characteristic . In a previous paper of ours, we defined and studied super-Hölder vectors in certain -linear representations of . In the present paper, we define and study super-Hölder vectors in certain -linear representations of a general -adic Lie group. We then consider certain -adic Lie extensions of a -adic field , and compute the super-Hölder vectors in the tilt of . We show that these super-Hölder vectors are the perfection of the field of norms of . By specializing to the case of a Lubin–Tate extension, we are able to recover inside the -adic completion of its perfection, seen as a valued -vector space endowed with the action of given by the endomorphisms of the corresponding Lubin–Tate group.
{"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}
Let be a number field, and a finite set of nonarchimedean places of , and write for the group of -units of . A famous theorem of Siegel asserts that the -unit equation , with , , has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over with good reduction outside . Now instead of a number field, let which denotes the -cyclotomic extension of . We show that the -unit equation , with , , has infinitely many solutions for , where consists only of the totally ramified prime above
{"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}
Pub Date : 2024-10-21DOI: 10.2140/ant.2024.18.2133
Anand Patel, Eric Riedl, Dennis Tseng
We study the variation of linear sections of hypersurfaces in . 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 hypersurface in varies maximally for . In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to -planes in projective space and to free rational curves on arbitrary varieties.
我们研究ℙn 中超曲面线段的变化。我们完整地分类了所有线段在模量上没有最大变化的平面曲线（必须是奇异曲线）。在更高维度上，我们证明了ℙn 中任何光滑度数为 d 的超曲面的超平面截面族在 d≥n+ 3 时变化最大。在此过程中，我们将关于投影空间中直线的经典格拉尔特-米利希定理推广到投影空间中的 k 平面和任意品种上的自由有理曲线。
{"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}
Pub Date : 2024-10-21DOI: 10.2140/ant.2024.18.2157
Artem Lopatin, Alexandr N. Zubkov
We describe separating -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 -invariants of several copies of the algebra of octonions in case of a field of odd characteristic.
{"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}
Pub Date : 2024-10-21DOI: 10.2140/ant.2024.18.2247
Márton Erdélyi, Árpád Tóth
We study a family of exponential sums that arises in the study of expanding horospheres on . We prove an explicit version of general purity and find optimal bounds for these sums.
我们研究了在 GL n 上扩展角球研究中出现的指数和族。我们证明了一般纯度的显式版本，并找到了这些和的最优边界。
{"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}
Pub Date : 2024-10-21DOI: 10.2140/ant.2024.18.2179
Lang Mou
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.
{"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}
We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple -values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that -conjugate rooted tree maps are their antipodes.
我们证明有根树映射的图像构成了多 L 值评估映射内核的子空间。为了证明这一点，我们将钻石积定义为修正的谐波积，并描述了它的性质。我们还证明了τ-共轭有根树映射是它们的反节点。
{"title":"Rooted tree maps for multiple L-values from a perspective of harmonic algebras","authors":"Hideki Murahara, Tatsushi Tanaka, Noriko Wakabayashi","doi":"10.2140/ant.2024.18.2003","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2003","url":null,"abstract":"<p>We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>τ</mi></math>-conjugate rooted tree maps are their antipodes. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449537","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}
Pub Date : 2024-10-18DOI: 10.2140/ant.2024.18.2091
Youness Lamzouri
We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants . In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to , and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to modulo . Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols.
{"title":"The distribution of large quadratic character sums and applications","authors":"Youness Lamzouri","doi":"10.2140/ant.2024.18.2091","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2091","url":null,"abstract":"<p>We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo>|</mo><mi>d</mi><mo>|</mo><mo>≤</mo>\u0000<mi>x</mi></math>. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math>, and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math>. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449539","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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