Journal of Number Theory最新文献

{"title":"Equidistribution of Hecke orbits on the Picard group of definite Shimura curves","authors":"Matias Alvarado,&nbsp;Patricio Pérez-Piña","doi":"10.1016/j.jnt.2025.05.002","DOIUrl":"10.1016/j.jnt.2025.05.002","url":null,"abstract":"<div><div>We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular Drinfeld modules of rank 2. Our approach is via the automorphic method, using bounds for coefficients of cuspidal automorphic forms of Drinfeld type as the main tool.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 715-725"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321677","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":"Almost primes between all squares","authors":"Adrian W. Dudek ,&nbsp;Daniel R. Johnston","doi":"10.1016/j.jnt.2025.05.009","DOIUrl":"10.1016/j.jnt.2025.05.009","url":null,"abstract":"<div><div>We prove that for all <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> there exists a number between <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> with at most 4 prime factors. This is the first result of this kind that holds for every <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> rather than just sufficiently large <em>n</em>. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 726-745"},"PeriodicalIF":0.6,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364442","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":"Fine Selmer groups of CM elliptic curves","authors":"Meng Fai Lim ,&nbsp;Jun Wang ,&nbsp;Dong Yan","doi":"10.1016/j.jnt.2025.05.012","DOIUrl":"10.1016/j.jnt.2025.05.012","url":null,"abstract":"<div><div>Let <em>E</em> be a CM elliptic curve defined over imaginary quadratic field <em>K</em> with good ordinary reduction at an odd prime <em>p</em>. We compute the sum of second Chern class of fine Selmer group and its involution. After specialization, we study certain properties of the zeros of the fine Selmer group over the anticyclotomic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension of <em>K</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 669-693"},"PeriodicalIF":0.6,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321681","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":"Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational j-invariant","authors":"Ivan Novak","doi":"10.1016/j.jnt.2025.05.007","DOIUrl":"10.1016/j.jnt.2025.05.007","url":null,"abstract":"<div><div>Let <em>E</em> be a non-CM elliptic curve with rational <em>j</em>-invariant. Mazur and Kenku determined all possible degrees of rational cyclic isogenies that <em>E</em> could have. One possible way of generalizing this would be to classify all possible degrees of cyclic isogenies defined over number fields of some fixed degree <span><math><mi>d</mi><mo>&gt;</mo><mn>1</mn></math></span>.</div><div>We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational <em>j</em>-invariant over number fields of degree <em>p</em>, where <em>p</em> is an odd prime. The analogous classification for quadratic number fields was done by Vukorepa, and this paper completes the classification in case when the degree of the number field is prime.</div><div>To prove the result, we make use of known results on Galois images of rational elliptic curves. In particular, when solving the cubic case, we use the database of images of 2-adic Galois representations which is due to Rouse and Zureick-Brown.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 694-714"},"PeriodicalIF":0.6,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321676","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 multiplicatively badly approximable vectors","authors":"Reynold Fregoli ,&nbsp;Dmitry Kleinbock","doi":"10.1016/j.jnt.2025.05.001","DOIUrl":"10.1016/j.jnt.2025.05.001","url":null,"abstract":"<div><div>Let <span><math><mo>〈</mo><mi>x</mi><mo>〉</mo></math></span> denote the distance from <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> to the set of integers <span><math><mi>Z</mi></math></span>. The Littlewood Conjecture states that for all pairs <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> the product <span><math><mi>q</mi><mo>〈</mo><mi>q</mi><mi>α</mi><mo>〉</mo><mo>〈</mo><mi>q</mi><mi>β</mi><mo>〉</mo></math></span> attains values arbitrarily close to 0 as <span><math><mi>q</mi><mo>∈</mo><mi>N</mi></math></span> tends to infinity. Badziahin showed that if a factor <span><math><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>⋅</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>q</mi></math></span> is added to the product, the same statement becomes false. In this paper, we generalise Badziahin's result to vectors <span><math><mi>α</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, replacing the function <span><math><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>⋅</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>q</mi></math></span> by <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>)</mo></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>q</mi></math></span> for any <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, and thereby obtaining a new proof in the case <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 570-621"},"PeriodicalIF":0.6,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321679","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":"A proof of a conjecture of Singh and Barman on hook length","authors":"Bing He,&nbsp;Shuming Liu","doi":"10.1016/j.jnt.2025.04.020","DOIUrl":"10.1016/j.jnt.2025.04.020","url":null,"abstract":"<div><div>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 the <em>t</em>-regular partitions of <em>n</em>. In this paper, we focus on inequalities on hook lengths, which Andrews, Ono, Singh and Barman etc. have studied previously. Applying inequalities on the modified Bessel function of the first kind and a modification of the circle method we prove that<span><span><span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>3</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>3</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span></span></span> holds for <span><math><mi>n</mi><mo>≥</mo><mn>28</mn></math></span>. This conforms a recent conjecture of Singh and Barman <span><span>[17]</span></span>. In addition, we also prove that<span><span><span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>4</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></span></span> holds for <span><math><mi>n</mi><mo>≥</mo><mn>82</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 353-379"},"PeriodicalIF":0.6,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263323","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":"Higher moments related to Dedekind zeta functions of non-normal fields","authors":"Jiong Yang ,&nbsp;Zhishan Yang","doi":"10.1016/j.jnt.2025.04.014","DOIUrl":"10.1016/j.jnt.2025.04.014","url":null,"abstract":"<div><div>Let <em>K</em> be a non-normal number field over <span><math><mi>Q</mi></math></span> with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>)</mo></math></span> the number of integer ideals in <em>K</em> of norm <span><math><mi>m</mi><mo>∈</mo><mi>Z</mi></math></span>. Let <em>L</em> be the Galois closure of <em>K</em> and assume that <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><mi>L</mi><mo>/</mo><mi>K</mi><mo>)</mo></math></span> is monomial. We obtain an asymptotic formula for the summation <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>m</mi><mo>≤</mo><mi>x</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mi>K</mi></mrow></msub><msup><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mi>l</mi></mrow></msup></math></span> for any <span><math><mi>l</mi><mo>≥</mo><mn>1</mn></math></span>. Moreover, in the dihedral case, we also obtain asymptotic formulas for the summation over a binary quadratic form. If <em>K</em> is a non-normal cubic field, this work refines previous works.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 547-569"},"PeriodicalIF":0.6,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272598","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":"Newspaces with nebentypus: An explicit dimension formula and classification of trivial newspaces","authors":"Erick Ross","doi":"10.1016/j.jnt.2025.04.003","DOIUrl":"10.1016/j.jnt.2025.04.003","url":null,"abstract":"<div><div>Consider <span><math><mi>N</mi><mo>≥</mo><mn>1</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, and <em>χ</em> a Dirichlet character modulo <em>N</em> such that <span><math><mi>χ</mi><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup></math></span>. For any bound <em>B</em>, one can show that <span><math><mi>dim</mi><mo>⁡</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><mi>χ</mi><mo>)</mo><mo>≤</mo><mi>B</mi></math></span> for only finitely many triples <span><math><mo>(</mo><mi>N</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>χ</mi><mo>)</mo></math></span>. It turns out that this property does not extend to the newspace; there exists an infinite family of triples <span><math><mo>(</mo><mi>N</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>χ</mi><mo>)</mo></math></span> for which <span><math><mi>dim</mi><mo>⁡</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>new</mtext></mrow></msubsup><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><mi>χ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. However, we classify this case entirely. We also show that excluding the infinite family for which <span><math><mi>dim</mi><mo>⁡</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>new</mtext></mrow></msubsup><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><mi>χ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, <span><math><mi>dim</mi><mo>⁡</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>new</mtext></mrow></msubsup><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><mi>χ</mi><mo>)</mo><mo>≤</mo><mi>B</mi></math></span> for only finitely many triples <span><math><mo>(</mo><mi>N</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>χ</mi><mo>)</mo></math></span>. In order to show these results, we derive an explicit dimension formula for the newspace <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow><mrow><mtext>new</mtext></mrow></msubsup><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><mi>χ</mi><mo>)</mo></math></span>. We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that <span><math><mi>dim</mi><mo>⁡</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mtext>new</mtext></mrow></msubsup><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>)</mo></math></span> takes on all possible non-negative integers.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 317-352"},"PeriodicalIF":0.6,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263279","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":"Brauer relations, isogenies and parities of ranks","authors":"Alexandros Konstantinou","doi":"10.1016/j.jnt.2025.04.016","DOIUrl":"10.1016/j.jnt.2025.04.016","url":null,"abstract":"<div><div>In this paper, we present applications of pseudo Brauer relations and their regulator constants in the study of isogenies and parities of Selmer ranks of Jacobians. In particular, we revisit and reconstruct a diverse array of classical isogenies in a uniform way and derive local formulae for Selmer rank parities, drawing from an extensive body of literature. These include local expressions found in the works of Birch–Cassels (isogenies between elliptic curves), Mazur–Rubin (dihedral extensions), Coates–Fukaya–Kato–Sujatha (<em>p</em>-power degree isogenies) and Kramer (quadratic twists of elliptic curves).</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 482-509"},"PeriodicalIF":0.6,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272595","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":"Intersection of orbits for polynomials in characteristic p","authors":"Simone Coccia ,&nbsp;Dragos Ghioca ,&nbsp;Jungin Lee ,&nbsp;GyeongHyeon Nam","doi":"10.1016/j.jnt.2025.04.019","DOIUrl":"10.1016/j.jnt.2025.04.019","url":null,"abstract":"<div><div>In <span><span>[GTZ08]</span></span>, <span><span>[GTZ12]</span></span>, the following result was established: given polynomials <span><math><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>C</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> of degrees larger than 1, if there exist <span><math><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>C</mi></math></span> such that their corresponding orbits <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>α</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>β</mi><mo>)</mo></math></span> (under the action of <em>f</em>, respectively of <em>g</em>) intersect in infinitely many points, then <em>f</em> and <em>g</em> must share a common iterate, i.e., <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>=</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for some <span><math><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. If one replaces <span><math><mi>C</mi></math></span> with a field <em>K</em> of characteristic <em>p</em>, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials <em>f</em> and <em>g</em> which admit two orbits with infinite intersection over a field of characteristic <em>p</em>. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 112-127"},"PeriodicalIF":0.6,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144243506","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}