Pub Date : 2024-06-21DOI: 10.1016/j.jnt.2024.05.001
Gurinder Singh, Rupam Barman
<div><p>In this article, we study hook lengths of ordinary partitions and <em>t</em>-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in 2-regular partitions. For a positive integer <em>k</em>, let <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of hooks of length <em>k</em> in all the partitions of <em>n</em>. We prove that <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>n</mi><mo>≠</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>; and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>1</mn></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. For integers <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, let <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of hooks of length <em>k</em> in all the <em>t</em>-regular partitions of <em>n</em>. We find generating functions of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for certain values of <em>t</em> and <em>k</em>. Exploring hook length biases for <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>></mo><mn>4</mn></math></span>, whereas <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>. We also propose some conjectures on biases among <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</m
本文研究普通分区和 t-regular 分区的钩长。我们建立了普通分区的钩长偏差，并在此基础上发现了 2-regular 分区中一些有趣的钩长偏差。对于正整数 k，让 p(k)(n) 表示 n 的所有分区中长度为 k 的钩码数。我们证明，对于所有 n≥0 和 n≠k+1 的情况，p(k)(n)≥p(k+1)(n)；对于 k≥2 的情况，p(k)(k+1)-p(k+1)(k+1)=-1。对于整数 t≥2 和 k≥1，让 bt,k(n)表示 n 的所有 t 规则分区中长度为 k 的钩子数。我们发现 bt,k(n)在某些 t 和 k 值下的生成函数。在探索 bt,k(n)的钩码长度偏差时，我们发现在某些情况下偏差与普通分区的偏差相反。我们证明了对于所有 n>4 b2,2(n)≥b2,1(n)，而对于所有 n≥0 b2,2(n)≥b2,3(n)。我们还提出了一些关于 bt,k(n) 偏差的猜想。
{"title":"Hook length biases in ordinary and t-regular partitions","authors":"Gurinder Singh, Rupam Barman","doi":"10.1016/j.jnt.2024.05.001","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.05.001","url":null,"abstract":"<div><p>In this article, we study hook lengths of ordinary partitions and <em>t</em>-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in 2-regular partitions. For a positive integer <em>k</em>, let <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of hooks of length <em>k</em> in all the partitions of <em>n</em>. We prove that <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>n</mi><mo>≠</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>; and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>1</mn></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. For integers <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, let <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of hooks of length <em>k</em> in all the <em>t</em>-regular partitions of <em>n</em>. We find generating functions of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for certain values of <em>t</em> and <em>k</em>. Exploring hook length biases for <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>></mo><mn>4</mn></math></span>, whereas <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>. We also propose some conjectures on biases among <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</m","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 41-58"},"PeriodicalIF":0.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-05DOI: 10.1016/j.jnt.2024.04.016
Thomas Brazelton , Joshua Harrington , Matthew Litman , Tony W.H. Wong
Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.
给定一个具有积分系数的多项式，我们可以探究它在素数 p 的调制下在其图像中可能的残差。在本文中，我们提供了奇特征有限域上任意阶狄克森多项式映像中不同残差之和的封闭形式，并证明了值集大小的完整特征。我们的结果为无界度多项式族的此类和提供了第一个非难分类。
{"title":"Residue sums of Dickson polynomials over finite fields","authors":"Thomas Brazelton , Joshua Harrington , Matthew Litman , Tony W.H. Wong","doi":"10.1016/j.jnt.2024.04.016","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.04.016","url":null,"abstract":"<div><p>Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime <em>p</em>. The sum over the distinct residues can sometimes be computed independent of the prime <em>p</em>; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 1-26"},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001240/pdfft?md5=f1a2e3015f4f9442190153e6f02f006d&pid=1-s2.0-S0022314X24001240-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.
{"title":"The ring of finite algebraic numbers and its application to the law of decomposition of primes","authors":"Julian Rosen , Yoshihiro Takeyama , Koji Tasaka , Shuji Yamamoto","doi":"10.1016/j.jnt.2024.04.003","DOIUrl":"10.1016/j.jnt.2024.04.003","url":null,"abstract":"<div><p>In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 335-365"},"PeriodicalIF":0.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Corrigendum to “On certain kernel functions and shifted convolution sums” [J. Number Theory 258 (2024) 414–450]","authors":"Kampamolla Venkatasubbareddy, Ayyadurai Sankaranarayanan","doi":"10.1016/j.jnt.2024.04.006","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.04.006","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 577-578"},"PeriodicalIF":0.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001045/pdfft?md5=7a793668b8ba10382a378b02c64a512a&pid=1-s2.0-S0022314X24001045-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141077971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.1016/j.jnt.2024.03.022
Barry Mazur , Karl Rubin , Alexandra Shlapentokh
{"title":"Corrigendum to “Existential definability and diophantine stability” [J. Number Theory 254 (2024) 1–64]","authors":"Barry Mazur , Karl Rubin , Alexandra Shlapentokh","doi":"10.1016/j.jnt.2024.03.022","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.03.022","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 539-540"},"PeriodicalIF":0.7,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24000970/pdfft?md5=4787a79a63b3819514bdc5118910efd9&pid=1-s2.0-S0022314X24000970-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141073460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.1016/j.jnt.2024.04.011
Shamik Das
In this paper, we establish a crucial requirement for a number of the form n, having two prime factors p and q such that , to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field when n is congruent.
在本文中，我们建立了一个关键的条件，即一个 n 形式的数，有两个质因数 p 和 q，且 (p,q)≡(1,3)(mod8), 才能被称为全等数。具体地说，我们提出了当 n 为全等数时，虚二次域 Q(-2pq) 的类数的 2 部分的 modulo 16 全等关系。
{"title":"Required condition for a congruent number: pq with primes p ≡ 1 (mod 8) and q ≡ 3 (mod 8)","authors":"Shamik Das","doi":"10.1016/j.jnt.2024.04.011","DOIUrl":"10.1016/j.jnt.2024.04.011","url":null,"abstract":"<div><p>In this paper, we establish a crucial requirement for a number of the form <em>n</em>, having two prime factors <em>p</em> and <em>q</em> such that <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>≡</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>8</mn><mo>)</mo></math></span>, to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>2</mn><mi>p</mi><mi>q</mi></mrow></msqrt><mo>)</mo></math></span> when <em>n</em> is congruent.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 139-152"},"PeriodicalIF":0.7,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141137962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.1016/j.jnt.2024.04.005
Manuel K.-H. Müller
For an isotropic subgroup H of a discriminant form D there exists a lift from modular forms for the Weil representation of the discriminant form to modular forms for the Weil representation of D. We determine a set of discriminant forms such that all modular forms for any discriminant form are induced from the discriminant forms in this set. Furthermore for any discriminant form in this set there exist modular forms that are not induced from smaller discriminant forms.
对于判别式 D 的各向同性子群 H，存在着从判别式 H⊥/H 的 Weil 表示的模块形式到 D 的 Weil 表示的模块形式的提升。此外，对于这个集合中的任何判别式，都存在不是从更小的判别式诱导出来的模块形式。
{"title":"Modular forms for the Weil representation induced from isotropic subgroups","authors":"Manuel K.-H. Müller","doi":"10.1016/j.jnt.2024.04.005","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.04.005","url":null,"abstract":"<div><p>For an isotropic subgroup <em>H</em> of a discriminant form <em>D</em> there exists a lift from modular forms for the Weil representation of the discriminant form <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>/</mo><mi>H</mi></math></span> to modular forms for the Weil representation of <em>D</em>. We determine a set of discriminant forms such that all modular forms for any discriminant form are induced from the discriminant forms in this set. Furthermore for any discriminant form in this set there exist modular forms that are not induced from smaller discriminant forms.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 206-233"},"PeriodicalIF":0.7,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001082/pdfft?md5=b8470837b07e5d1a04073db4dbf9f70c&pid=1-s2.0-S0022314X24001082-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141163696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-20DOI: 10.1016/j.jnt.2024.04.015
Karim Boulajhaf, Ali Mouhib
Let k be a real quadratic number field, and its cyclotomic -extension. We study the cyclicity of the Galois group over of the maximal abelian unramified 2-extension, in which all 2-adic primes of split completely. As consequence, we determine the complete list of real quadratic number fields for which is cyclic.
When is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.
{"title":"Cyclicity of the 2-decomposed unramified Iwasawa module","authors":"Karim Boulajhaf, Ali Mouhib","doi":"10.1016/j.jnt.2024.04.015","DOIUrl":"10.1016/j.jnt.2024.04.015","url":null,"abstract":"<div><p>Let <em>k</em> be a real quadratic number field, and <span><math><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> its cyclotomic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-extension. We study the cyclicity of the Galois group <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> over <span><math><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> of the maximal abelian unramified 2-extension, in which all 2-adic primes of <span><math><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> split completely. As consequence, we determine the complete list of real quadratic number fields for which <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> is cyclic.</p><p>When <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 234-254"},"PeriodicalIF":0.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141141473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-20DOI: 10.1016/j.jnt.2024.04.013
Boris Springborn
We classify and enumerate all rational numbers with approximation constant at least using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.
{"title":"The worst approximable rational numbers","authors":"Boris Springborn","doi":"10.1016/j.jnt.2024.04.013","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.04.013","url":null,"abstract":"<div><p>We classify and enumerate all rational numbers with approximation constant at least <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of <em>Markov fractions</em> whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of <em>companions</em>, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 153-205"},"PeriodicalIF":0.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001148/pdfft?md5=05527a34f5adb10106a0ae68575e41cc&pid=1-s2.0-S0022314X24001148-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141163695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-20DOI: 10.1016/j.jnt.2024.04.002
Pierre Colmez , Shanwen Wang
We extend the dictionary between Fontaine rings and p-adic functionnal analysis, and we give a refinement of the p-adic local Langlands correspondence for principal series representations of .
{"title":"Fonctions d'une variable p-adique et représentations de GL2(Qp)","authors":"Pierre Colmez , Shanwen Wang","doi":"10.1016/j.jnt.2024.04.002","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.04.002","url":null,"abstract":"<div><p>We extend the dictionary between Fontaine rings and <em>p</em>-adic functionnal analysis, and we give a refinement of the <em>p</em>-adic local Langlands correspondence for principal series representations of <span><math><msub><mrow><mtext>GL</mtext></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"263 ","pages":"Pages 1-23"},"PeriodicalIF":0.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141095666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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