Journal of Number Theory最新文献

{"title":"Herr complex of (φ,τ)-modules","authors":"Luming Zhao","doi":"10.1016/j.jnt.2024.10.012","DOIUrl":"10.1016/j.jnt.2024.10.012","url":null,"abstract":"<div><div>Let <em>p</em> be an odd prime number and <em>K</em> a mixed characteristic complete discrete valuation field with perfect residue field of characteristic <em>p</em>. We construct a three-term complex, defined in terms of the <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>-module of a <em>p</em>-adic representation and prove its homology is isomorphic to the Galois cohomology of the representation. We further show that our complex is quasi-isomorphic to the four-term complex constructed by Tavares Ribeiro, providing an alternative proof of our result. As an application, we describe Galois cohomology of the Tate module associated to a <em>p</em>-divisible group in terms of corresponding Breuil-Kisin modules.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"271 ","pages":"Pages 66-98"},"PeriodicalIF":0.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128941","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":"Second JNT Biennial Conference 2022","authors":"","doi":"10.1016/j.jnt.2024.11.001","DOIUrl":"10.1016/j.jnt.2024.11.001","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"270 ","pages":"Pages 1-3"},"PeriodicalIF":0.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551364","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":"Preface to “Proceedings of the 2nd JNT Biennial Conference, 2022”","authors":"Dorian Goldfeld,&nbsp;Federico Pellarin,&nbsp;Lejla Smajlovic","doi":"10.1016/j.jnt.2024.11.002","DOIUrl":"10.1016/j.jnt.2024.11.002","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"270 ","pages":"Pages 4-5"},"PeriodicalIF":0.6,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551365","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 maximal hyperelliptic curves related to Chebyshev polynomials” [J. Number Theory 203 (2019) 276–293]","authors":"Saeed Tafazolian ,&nbsp;Jaap Top","doi":"10.1016/j.jnt.2024.10.011","DOIUrl":"10.1016/j.jnt.2024.10.011","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 427-428"},"PeriodicalIF":0.6,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744084","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":"Representation functions in the set of natural numbers","authors":"Shi-Qiang Chen ,&nbsp;Csaba Sándor ,&nbsp;Quan-Hui Yang","doi":"10.1016/j.jnt.2024.10.007","DOIUrl":"10.1016/j.jnt.2024.10.007","url":null,"abstract":"<div><div>Let <span><math><mi>N</mi></math></span> be the set of all nonnegative integers. For <span><math><mi>S</mi><mo>⊆</mo><mi>N</mi></math></span> and <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>, let <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of solutions of the equation <span><math><mi>n</mi><mo>=</mo><mi>s</mi><mo>+</mo><msup><mrow><mi>s</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, <span><math><mi>s</mi><mo>,</mo><msup><mrow><mi>s</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>S</mi></math></span>, <span><math><mi>s</mi><mo>&lt;</mo><msup><mrow><mi>s</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. In this paper, we determine the structure of all sets <em>A</em> and <em>B</em> such that <span><math><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mi>N</mi><mo>∖</mo><mo>{</mo><mi>r</mi><mo>+</mo><mi>m</mi><mi>k</mi><mo>:</mo><mi>k</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span>, <span><math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mo>∅</mo></math></span> and <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for every positive integer <em>n</em>, where <em>m</em> and <em>r</em> are two integers with <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 465-495"},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133639","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":"Sumset problem on dilated sets of integers","authors":"Sandeep Singh ,&nbsp;Ramandeep Kaur ,&nbsp;Mamta Verma","doi":"10.1016/j.jnt.2024.10.006","DOIUrl":"10.1016/j.jnt.2024.10.006","url":null,"abstract":"<div><div>Let <em>A</em> be a non-empty finite set of integers. For integers <em>m</em> and <em>k</em>, let <span><math><mi>m</mi><mo>⋅</mo><mi>A</mi><mo>+</mo><mi>k</mi><mo>⋅</mo><mi>A</mi><mo>=</mo><mo>{</mo><mi>m</mi><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>k</mi><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>A</mi><mo>}</mo></math></span>. For <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span> and an odd prime <em>k</em> such that <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>&gt;</mo><mn>8</mn><msup><mrow><mi>k</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, Hamidoune et al. <span><span>[6]</span></span> proved that <span><math><mo>|</mo><mn>2</mn><mo>⋅</mo><mi>A</mi><mo>+</mo><mi>k</mi><mo>⋅</mo><mi>A</mi><mo>|</mo><mo>≥</mo><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>k</mi><mo>+</mo><mn>2</mn></math></span>. Ljujic <span><span>[7]</span></span> extended this result and obtained the same bound for <em>k</em> to be a power of an odd prime and product of two distinct odd primes. Balog et al. <span><span>[1]</span></span> proved that <span><math><mo>|</mo><mi>p</mi><mo>⋅</mo><mi>A</mi><mo>+</mo><mi>q</mi><mo>⋅</mo><mi>A</mi><mo>|</mo><mo>≥</mo><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>−</mo><msup><mrow><mo>(</mo><mi>p</mi><mi>q</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>)</mo><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msup></math></span>, where <span><math><mi>p</mi><mo>&lt;</mo><mi>q</mi></math></span> are relatively primes. In this article, for any odd values of <em>k</em> and under some certain conditions on set <em>A</em>, we obtain that <span><math><mo>|</mo><mn>2</mn><mo>⋅</mo><mi>A</mi><mo>+</mo><mi>k</mi><mo>⋅</mo><mi>A</mi><mo>|</mo><mo>≥</mo><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>−</mo><mn>2</mn><mi>k</mi><mo>|</mo><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>|</mo></math></span>, where <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> is the projection of <em>A</em> in <span><math><mi>Z</mi><mo>/</mo><mi>k</mi><mi>Z</mi></math></span>. This obtained bound is better than the bound given by Balog et al. We also generalize this bound for <span><math><mo>|</mo><mi>p</mi><mo>⋅</mo><mi>A</mi><mo>+</mo><mi>k</mi><mo>⋅</mo><mi>A</mi><mo>|</mo></math></span>, where <em>p</em> is any odd prime and <em>k</em> be an odd positive integer with <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 429-439"},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133637","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":"Period of the Ikeda-Miyawaki lift","authors":"Tomoyoshi Ibukiyama ,&nbsp;Hidenori Katsurada ,&nbsp;Hisashi Kojima","doi":"10.1016/j.jnt.2024.09.014","DOIUrl":"10.1016/j.jnt.2024.09.014","url":null,"abstract":"<div><div>In this paper, first we give a weak version of Ikeda's conjecture on the period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously in some cases.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 341-369"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744086","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":"Rational configuration problems and a family of curves","authors":"Jonathan Love","doi":"10.1016/j.jnt.2024.09.008","DOIUrl":"10.1016/j.jnt.2024.09.008","url":null,"abstract":"<div><div>Given <figure><img></figure>, we consider the number of rational points on the genus one curve<span><span><span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><mi>b</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>c</mi><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></math></span></span></span> We prove that the set of <em>η</em> for which <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo><mo>≠</mo><mo>∅</mo></math></span> has density zero, and that if a rational point <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>∈</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> exists, then <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> is infinite unless a certain explicit polynomial in <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> vanishes.</div><div>Curves of the form <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub></math></span> naturally occur in the study of configurations of points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> passes through a rational point on the unit circle, then it contains a dense set of points <em>P</em> such that the distances from <em>P</em> to each of the three points <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> are all rational. We also prove some results regarding whether a rational number can be expressed as a sum or product of slopes of rational right triangles.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 370-396"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744087","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}

{"title":"On gamma factors of Rankin–Selberg integrals for U2ℓ × ResE/FGLn","authors":"Kazuki Morimoto","doi":"10.1016/j.jnt.2024.09.013","DOIUrl":"10.1016/j.jnt.2024.09.013","url":null,"abstract":"<div><div>In this paper, we prove the fundamental properties of gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of generic representations <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math></span> for a local field <em>F</em> of characteristic zero and a quadratic extension <em>E</em> of <em>F</em>. We also prove similar results for pairs of generic representations <span><math><mo>(</mo><mi>π</mi><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>×</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span>. As a corollary, we prove that the gamma factors arising from Langlands–Shahidi method and our gamma factors coincide.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 203-246"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744090","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":"On the height of some generators of Galois extensions with big Galois group","authors":"Jonathan Jenvrin","doi":"10.1016/j.jnt.2024.10.004","DOIUrl":"10.1016/j.jnt.2024.10.004","url":null,"abstract":"<div><div>We study the height of generators of Galois extensions of the rationals having the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height tends to infinity as <em>n</em> increases. This provides an analogue of a result by Amoroso, originally established for the symmetric group.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 78-105"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744205","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}