Journal of Number Theory最新文献

{"title":"Ring class fields and a result of Hasse","authors":"","doi":"10.1016/j.jnt.2024.07.001","DOIUrl":"10.1016/j.jnt.2024.07.001","url":null,"abstract":"<div><p>For squarefree <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn></math></span>, let <em>M</em> denote the ring class field for the order <span><math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mn>3</mn><mi>d</mi></mrow></msqrt><mo>]</mo></math></span> in <span><math><mi>F</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>. Hasse proved that 3 divides the class number of <em>F</em> if and only if there exists a cubic extension <em>E</em> of <span><math><mi>Q</mi></math></span> such that <em>E</em> and <em>F</em> have the same discriminant. Define the real cube roots <span><math><mi>v</mi><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span>, where <span><math><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt></math></span> is the fundamental unit in <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>. We prove that <em>E</em> can be taken as <span><math><mi>Q</mi><mo>(</mo><mi>v</mi><mo>+</mo><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span> if and only if <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span>. As byproducts of the proof, we give explicit congruences for <em>a</em> and <em>b</em> which hold if and only if <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span>, and we also show that the norm of the relative discriminant of <span><math><mi>F</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>/</mo><mi>F</mi></math></span> lies in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>}</mo></math></span> or <span><math><mo>{</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>8</mn></mrow></msup><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>18</mn></mrow></msup><mo>}</mo></math></span> according as <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span> or <span><math><mi>v</mi><mo>∉</mo><mi>M</mi></math></span>. We then prove that <em>v</em> is always in the ring class field for the order <span><math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mn>27</mn><mi>d</mi></mrow></msqrt><mo>]</mo></math></span> in <em>F</em>. Some of the results above are extended for subsets of <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span> properly containing the fundamental units <span><math><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt></math></span>.</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/S0022314X24001677/pdfft?md5=4a76de3ef7096a558707691b3467bc3b&pid=1-s2.0-S0022314X24001677-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012867","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":"The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II","authors":"","doi":"10.1016/j.jnt.2024.07.013","DOIUrl":"10.1016/j.jnt.2024.07.013","url":null,"abstract":"<div><p>Let <em>E</em> and <span><math><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> be 2-isogenous elliptic curves over <strong>Q</strong>. Following <span><span>[6]</span></span>, we call a prime of good reduction <em>p anomalous</em> if <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>≃</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> but <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo><mo>≄</mo><msup><mrow><mi>E</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math></span>. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.</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/S0022314X24001720/pdfft?md5=f8f53d9d54ebb568a03018d889d8244b&pid=1-s2.0-S0022314X24001720-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083985","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":"Progress towards a conjecture of S.W. Graham","authors":"","doi":"10.1016/j.jnt.2024.07.009","DOIUrl":"10.1016/j.jnt.2024.07.009","url":null,"abstract":"<div><p>This article describes progress towards a conjecture of S.W. Graham. He conjectured that the number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of Carmichael numbers up to <em>X</em> with three prime factors is <span><math><mo>≤</mo><msqrt><mrow><mi>X</mi></mrow></msqrt></math></span> for all <span><math><mi>X</mi><mo>≥</mo><mn>1</mn></math></span>. He showed that his conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>&gt;</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>126</mn></mrow></msup></math></span>. In this article, it is shown that the conjecture is true for <span><math><mi>X</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>24</mn></mrow></msup></math></span> and <span><math><mi>X</mi><mo>&gt;</mo><mn>2</mn><mo>⁎</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>40</mn></mrow></msup></math></span>. In both cases, analytical methods establish the conjecture for large <em>X</em> and tables of Carmichael numbers are used for small <em>X</em>.</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/S0022314X24001756/pdfft?md5=a96d01f7aa1e622c98ed012747b85804&pid=1-s2.0-S0022314X24001756-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097231","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":"Some computational results on a conjecture of de Polignac about numbers of the form p + 2k","authors":"","doi":"10.1016/j.jnt.2024.07.004","DOIUrl":"10.1016/j.jnt.2024.07.004","url":null,"abstract":"<div><p>Let <span><math><mi>U</mi></math></span> be the set of positive odd numbers that can not be written in the form <span><math><mi>p</mi><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. Recently, by analyzing possible prime divisors of <em>b</em>, Chen proved <span><math><mi>b</mi><mo>≥</mo><mn>11184810</mn></math></span> and <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>≥</mo><mn>7</mn></math></span> if an arithmetic progression <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>b</mi><mo>)</mo></math></span> is in <span><math><mi>U</mi></math></span>, with <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>=</mo><mn>7</mn></math></span> if and only if <span><math><mi>b</mi><mo>=</mo><mn>11184810</mn></math></span>, where <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the number of distinct prime divisors of <em>n</em>. In this paper, we take a computational approach to prove <span><math><mi>b</mi><mo>≥</mo><mn>11184810</mn></math></span> and provide all possible values of <em>a</em> if <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>11184810</mn><mo>)</mo></math></span> is in <span><math><mi>U</mi></math></span>. Moreover, we explicitly construct nontrivial arithmetic progressions <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>b</mi><mo>)</mo></math></span> in <span><math><mi>U</mi></math></span> with <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>=</mo><mn>8</mn></math></span>, 9, 10, or 11, and provide potential nontrivial arithmetic progressions <span><math><mi>a</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>b</mi><mo>)</mo></math></span> in <span><math><mi>U</mi></math></span> such that <span><math><mi>ω</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>=</mo><mi>s</mi></math></span> for any fixed <span><math><mi>s</mi><mo>≥</mo><mn>12</mn></math></span>. Furthermore, we improve the upper bound estimate of numbers of the form <span><math><mi>p</mi><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span> by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.</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/S0022314X24001690/pdfft?md5=34d1d69eca5fb0f5a9878d3392dfc7c6&pid=1-s2.0-S0022314X24001690-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097312","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 Erdős covering systems in global function fields","authors":"","doi":"10.1016/j.jnt.2024.07.002","DOIUrl":"10.1016/j.jnt.2024.07.002","url":null,"abstract":"<div><p>A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most 10¹⁶. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to <span><math><mn>616</mn><mo>,</mo><mn>000</mn></math></span> by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity <em>s</em> in any global function field of genus <em>g</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for <span><math><mi>q</mi><mo>≥</mo><mo>(</mo><mn>1.14</mn><mo>+</mo><mn>0.16</mn><mi>g</mi><mo>)</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>6.5</mn><mo>+</mo><mn>0.97</mn><mi>g</mi></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. In particular, there is no covering system of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> with distinct moduli for <span><math><mi>q</mi><mo>≥</mo><mn>759</mn></math></span>.</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/S0022314X24001707/pdfft?md5=f62f3fbb627421563f5c92d4564888ee&pid=1-s2.0-S0022314X24001707-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097370","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":"Non-zero central values of Dirichlet twists of elliptic L-functions","authors":"","doi":"10.1016/j.jnt.2024.07.003","DOIUrl":"10.1016/j.jnt.2024.07.003","url":null,"abstract":"<div><p>We consider heuristic predictions for small non-zero algebraic central values of twists of the <em>L</em>-function of an elliptic curve <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> extended to chosen families of cyclic extensions of fixed degree.</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/S0022314X24001719/pdfft?md5=a0edf993c65449e6ef9e685a75b7c9ac&pid=1-s2.0-S0022314X24001719-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097371","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":"Kurokawa-Mizumoto congruence and differential operators on automorphic forms","authors":"","doi":"10.1016/j.jnt.2024.07.007","DOIUrl":"10.1016/j.jnt.2024.07.007","url":null,"abstract":"<div><p>We give sufficient conditions for the vector-valued Kurokawa-Mizumoto congruence related to the Klingen-Eisenstein series to hold. We also give a reinterpretation for differential operators on automorphic forms by the representation theory.</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/S0022314X24001689/pdfft?md5=e293d3f84e4efad68424f0d31bf2f6e3&pid=1-s2.0-S0022314X24001689-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076717","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":"Ichino periods for CM forms","authors":"","doi":"10.1016/j.jnt.2024.07.011","DOIUrl":"10.1016/j.jnt.2024.07.011","url":null,"abstract":"<div><p>In both local and global settings, we establish explicit relations between Ichino triple product period and Waldspurger toric periods for CM forms via the theta lifting and the see-saw principle.</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/S0022314X24001732/pdfft?md5=e4fad99bbbcdcee7db9508d12470d009&pid=1-s2.0-S0022314X24001732-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097365","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 Bh[1]-sets which are asymptotic bases of order 2h","authors":"","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}