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Residue sums of Dickson polynomials over finite fields 有限域上狄克森多项式的残差和
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-06-05 DOI: 10.1016/j.jnt.2024.04.016
Thomas Brazelton , Joshua Harrington , Matthew Litman , Tony W.H. Wong
{"title":"Residue sums of Dickson polynomials over finite fields","authors":"Thomas Brazelton ,&nbsp;Joshua Harrington ,&nbsp;Matthew Litman ,&nbsp;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":null,"pages":null},"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}
引用次数: 0
The ring of finite algebraic numbers and its application to the law of decomposition of primes 有限代数数环及其在素数分解定律中的应用
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-24 DOI: 10.1016/j.jnt.2024.04.003
Julian Rosen , Yoshihiro Takeyama , Koji Tasaka , Shuji Yamamoto
{"title":"The ring of finite algebraic numbers and its application to the law of decomposition of primes","authors":"Julian Rosen ,&nbsp;Yoshihiro Takeyama ,&nbsp;Koji Tasaka ,&nbsp;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":null,"pages":null},"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}
引用次数: 0
Corrigendum to “On certain kernel functions and shifted convolution sums” [J. Number Theory 258 (2024) 414–450] 关于某些核函数和移位卷积和》的更正 [J. Number Theory 258 (2024) 414-450]
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-22 DOI: 10.1016/j.jnt.2024.04.006
Kampamolla Venkatasubbareddy, Ayyadurai Sankaranarayanan
{"title":"Corrigendum to “On certain kernel functions and shifted convolution sums” [J. Number Theory 258 (2024) 414–450]","authors":"Kampamolla Venkatasubbareddy,&nbsp;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":null,"pages":null},"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}
引用次数: 0
Corrigendum to “Existential definability and diophantine stability” [J. Number Theory 254 (2024) 1–64] 对 "存在可定义性和二项稳定性 "的更正 [J. Number Theory 254 (2024) 1-64]
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-21 DOI: 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 ,&nbsp;Karl Rubin ,&nbsp;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":null,"pages":null},"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}
引用次数: 0
Required condition for a congruent number: pq with primes p ≡ 1 (mod 8) and q ≡ 3 (mod 8) 全等数的必要条件：pq 的素数 p≡1 (mod 8) 和 q≡3 (mod 8)
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-21 DOI: 10.1016/j.jnt.2024.04.011
Shamik Das
{"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":null,"pages":null},"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}
引用次数: 0
Modular forms for the Weil representation induced from isotropic subgroups 各向同性子群诱导的魏尔表示的模块形式
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-21 DOI: 10.1016/j.jnt.2024.04.005
Manuel K.-H. Müller
{"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":null,"pages":null},"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}
引用次数: 0
Cyclicity of the 2-decomposed unramified Iwasawa module 岩泽模块的 2 分解非ramified 循环性
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-20 DOI: 10.1016/j.jnt.2024.04.015
Karim Boulajhaf, Ali Mouhib
{"title":"Cyclicity of the 2-decomposed unramified Iwasawa module","authors":"Karim Boulajhaf,&nbsp;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":null,"pages":null},"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}
引用次数: 0
The worst approximable rational numbers 最差近似有理数
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-20 DOI: 10.1016/j.jnt.2024.04.013
Boris Springborn
{"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":null,"pages":null},"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}
引用次数: 0
Fonctions d'une variable p-adique et représentations de GL2(Qp) p 自变量函数和 GL2(Qp) 表示
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-20 DOI: 10.1016/j.jnt.2024.04.002
Pierre Colmez , Shanwen Wang
{"title":"Fonctions d'une variable p-adique et représentations de GL2(Qp)","authors":"Pierre Colmez ,&nbsp;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":null,"pages":null},"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}
引用次数: 0
Non-vanishing of multiple zeta values for higher genus curves over finite fields 有限域上高属曲线的多重zeta值不求和
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2024-05-17 DOI: 10.1016/j.jnt.2024.04.014
Daichi Matsuzuki
{"title":"Non-vanishing of multiple zeta values for higher genus curves over finite fields","authors":"Daichi Matsuzuki","doi":"10.1016/j.jnt.2024.04.014","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.04.014","url":null,"abstract":"<div><p>In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141084645","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}
引用次数: 0
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