Journal of Number Theory最新文献

{"title":"On Bh[1]-sets which are asymptotic bases of order 2h","authors":"Sándor Z. Kiss ,&nbsp;Csaba Sándor","doi":"10.1016/j.jnt.2024.07.006","DOIUrl":"10.1016/j.jnt.2024.07.006","url":null,"abstract":"<div><p>Let <span><math><mi>h</mi><mo>,</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be integers. A set <em>A</em> of positive integers is called asymptotic basis of order <em>k</em> if every large enough positive integer can be written as the sum of <em>k</em> terms from <em>A</em>. A set of positive integers <em>A</em> is said to be a <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>[</mo><mi>g</mi><mo>]</mo></math></span>-set if every positive integer can be written as the sum of <em>h</em> terms from <em>A</em> at most <em>g</em> different ways. In this paper we prove the existence of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>[</mo><mn>1</mn><mo>]</mo></math></span> sets which are asymptotic bases of order 2<em>h</em> by using probabilistic methods.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X2400177X/pdfft?md5=312356ef445f315e287739fcb2d6b0f7&pid=1-s2.0-S0022314X2400177X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162048","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":"Kato's epsilon conjecture for anticyclotomic CM deformations at inert primes","authors":"Ashay A. Burungale, Shinichi Kobayashi, Kazuto Ota, Seidai Yasuda","doi":"10.1016/j.jnt.2024.06.014","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.06.014","url":null,"abstract":"We present an explicit construction of Kato's local epsilon isomorphism for the anticyclotomic deformation of a Lubin-Tate formal group of height two by using Rubin's theory on local units in the anticyclotomic tower. We also prove Kato's global epsilon conjecture for the anticyclotomic deformation of a CM elliptic curve at an inert prime.","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185486","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":"2∞-Selmer rank parities via the Prym construction","authors":"Jordan Docking","doi":"10.1016/j.jnt.2024.06.009","DOIUrl":"10.1016/j.jnt.2024.06.009","url":null,"abstract":"<div><p>We derive a local formula for the parity of the <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-Selmer rank of Jacobians of curves of genus 2 or 3 which admit an unramified double cover. We give an explicit example to show how this local formula gives rank parity predictions against which the 2-parity conjecture may be tested. Our results yield applications to the parity conjecture for semistable curves of genus 3.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001525/pdfft?md5=19533e2a3ab48597f61bd70dc3f8df28&pid=1-s2.0-S0022314X24001525-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780781","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":"Class group and factorization in orders of a PID","authors":"Hyun Seung Choi","doi":"10.1016/j.jnt.2024.06.008","DOIUrl":"10.1016/j.jnt.2024.06.008","url":null,"abstract":"<div><p>In this paper, we study properties of factorization in orders of a PID via the computation of algebraic invariants that measure the failure of unique factorization. The focus is on the numerical semigroup rings over a finite field and the orders of imaginary quadratic fields with class number 1. We also give a complete description of the class group structure of those rings.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780789","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 number variance of sequences with small additive energy","authors":"Zonglin Li ,&nbsp;Nadav Yesha","doi":"10.1016/j.jnt.2024.06.006","DOIUrl":"10.1016/j.jnt.2024.06.006","url":null,"abstract":"<div><p>For a real-valued sequence <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span>, denote by <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>)</mo></math></span> the number of its first <em>N</em> fractional parts lying in a random interval of size <span><math><mi>ℓ</mi><mo>:</mo><mo>=</mo><mi>L</mi><mo>/</mo><mi>N</mi></math></span>, where <span><math><mi>L</mi><mo>=</mo><mi>o</mi><mo>(</mo><mi>N</mi><mo>)</mo></math></span> as <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span>. We study the variance of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>)</mo></math></span> (the number variance) for sequences of the form <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>α</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> is a sequence of distinct integers. We show that if the additive energy of the sequence <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> is bounded from above by <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn><mo>−</mo><mi>ε</mi></mrow></msup><mo>/</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for some <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span>, then for almost all <em>α</em>, the number variance is asymptotic to <em>L</em> (Poissonian number variance). This holds in particular for the sequence <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>α</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>d</mi><mo>≥</mo><mn>2</mn></math></span> whenever <span><math><mi>L</mi><mo>=</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span> with <span><math><mn>0</mn><mo>≤</mo><mi>β</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001513/pdfft?md5=37404fefcd835f751277ba8aa774bc81&pid=1-s2.0-S0022314X24001513-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780783","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":"A segment of Euler product associated to a certain Dirichlet series","authors":"Rajat Gupta ,&nbsp;Aditi Savalia","doi":"10.1016/j.jnt.2024.06.003","DOIUrl":"10.1016/j.jnt.2024.06.003","url":null,"abstract":"<div><p>In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>|</mo><mi>n</mi></mrow></msub><msup><mrow><mi>d</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>. We obtain an exact identity relating the Dirichlet series <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>−</mo><mi>α</mi><mo>)</mo></math></span> and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780787","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":"Density of power-free values of polynomials II","authors":"Kostadinka Lapkova ,&nbsp;Stanley Yao Xiao","doi":"10.1016/j.jnt.2024.06.010","DOIUrl":"10.1016/j.jnt.2024.06.010","url":null,"abstract":"<div><p>In this paper we prove that polynomials <span><math><mi>F</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> of degree <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, satisfying certain hypotheses, take on the expected density of <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-free values. This extends the authors' earlier result in <span><span>[14]</span></span> where a different method implied the similar statement for polynomials of degree <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001550/pdfft?md5=5679964f477441d43dd0509c9504b52e&pid=1-s2.0-S0022314X24001550-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780786","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 non-Zariski density of (D,S)-integral points in forward orbits and the Subspace Theorem","authors":"Nathan Grieve ,&nbsp;Chatchai Noytaptim","doi":"10.1016/j.jnt.2024.06.005","DOIUrl":"10.1016/j.jnt.2024.06.005","url":null,"abstract":"<div><p>Working over a base number field <strong>K</strong>, we study the attractive question of Zariski non-density for <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>-integral points in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> the forward <em>f</em>-orbit of a rational point <span><math><mi>x</mi><mo>∈</mo><mi>X</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. Here, <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> is a regular surjective self-map for <em>X</em> a geometrically irreducible projective variety over <strong>K</strong>. Given a non-zero and effective <em>f</em>-quasi-polarizable Cartier divisor <em>D</em> on <em>X</em> and defined over <strong>K</strong>, our main result gives a sufficient condition, that is formulated in terms of the <em>f</em>-dynamics of <em>D</em>, for non-Zariski density of certain dynamically defined subsets of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. For the case of <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>-integral points, this result gives a sufficient condition for non-Zariski density of integral points in <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Our approach expands on that of Yasufuku, <span><span>[13]</span></span>, building on earlier work of Silverman <span><span>[11]</span></span>. Our main result gives an unconditional form of the main results of <span><span>[13]</span></span>; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in <span><span>[10]</span></span> and expanded upon in <span><span>[3]</span></span> and <span><span>[6]</span></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001495/pdfft?md5=b3dd7c5b16ab793f55d50824e16a3394&pid=1-s2.0-S0022314X24001495-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785716","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":"A twisted additive divisor problem","authors":"Alex Cowan","doi":"10.1016/j.jnt.2024.06.007","DOIUrl":"10.1016/j.jnt.2024.06.007","url":null,"abstract":"<div><p>We give asymptotics for shifted convolutions of the form<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>&lt;</mo><mi>X</mi></mrow></munder><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>u</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>χ</mi><mo>)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>v</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow></msup></mrow></mfrac></math></span></span></span> for nonzero complex numbers <span><math><mi>u</mi><mo>,</mo><mi>v</mi></math></span> and nontrivial Dirichlet characters <span><math><mi>χ</mi><mo>,</mo><mi>ψ</mi></math></span>. We use the technique of <em>automorphic regularization</em> to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001586/pdfft?md5=03e4c4b87cd43372c6c4156f7d76d43a&pid=1-s2.0-S0022314X24001586-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780782","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":"Sums of coefficients of general L-functions over arithmetic progressions and applications","authors":"Dan Wang","doi":"10.1016/j.jnt.2024.06.011","DOIUrl":"10.1016/j.jnt.2024.06.011","url":null,"abstract":"<div><p>In this paper, we study the asymptotic distribution of coefficients of general <em>L</em>-functions over arithmetic progressions without the Ramanujan conjecture. As an application, we consider the high mean of Fourier coefficients of holomorphic forms or Maass forms for <span><math><mi>Γ</mi><mo>=</mo><mrow><mi>SL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>Z</mi><mo>)</mo></math></span> over arithmetic progressions, and improve the results of Jiang and Lü <span><span>[10]</span></span>. Our new results remove the restriction to prime module and improve the interval length of module <em>q</em>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780784","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}