Journal of Number Theory最新文献

{"title":"Counting rational points on the sphere with bounded denominator","authors":"Dubi Kelmer","doi":"10.1016/j.jnt.2025.11.015","DOIUrl":"10.1016/j.jnt.2025.11.015","url":null,"abstract":"<div><div>We give an optimal bound for the remainder when counting the number of rational points on the <em>n</em>-dimensional sphere with bounded denominator for any <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 174-191"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884097","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":"Squares in arithmetic progression over certain non-primitive quartic number fields","authors":"Enrique González–Jiménez ,&nbsp;Nguyen Xuan Tho","doi":"10.1016/j.jnt.2025.11.005","DOIUrl":"10.1016/j.jnt.2025.11.005","url":null,"abstract":"<div><div>Let <em>D</em> be a square-free integer. Under certain conditions on <em>D</em>, we characterize non-constant arithmetic progressions of squares over quadratic extensions of <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>D</mi></mrow></msqrt><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 13-30"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685021","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":"Counting ideals in abelian number fields","authors":"Alessandro Languasco ,&nbsp;Rashi Lunia ,&nbsp;Pieter Moree","doi":"10.1016/j.jnt.2025.10.012","DOIUrl":"10.1016/j.jnt.2025.10.012","url":null,"abstract":"<div><div>Already Dedekind and Weber considered the problem of counting integral ideals of norm at most <em>x</em> in a given number field <em>K</em>. Here we improve on the existing results in case <span><math><mi>K</mi><mo>/</mo><mi>Q</mi></math></span> is abelian and has degree at least four. For these fields, we obtain as a consequence an improvement of the available results on counting pairs of coprime ideals each having norm at most <em>x</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 1-12"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625095","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 irrationals with Lagrange value exactly 3","authors":"Zhe Cao ,&nbsp;Harold Erazo ,&nbsp;Carlos Gustavo Moreira","doi":"10.1016/j.jnt.2025.11.006","DOIUrl":"10.1016/j.jnt.2025.11.006","url":null,"abstract":"<div><h3>Text</h3><div>For <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span>, let <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> denote the set of <span><math><mi>x</mi><mo>∈</mo><mi>R</mi><mo>﹨</mo><mi>Q</mi></math></span> such that <span><math><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac><mo>|</mo></mrow><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>c</mi><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span> has only finitely many rational solutions <span><math><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac></math></span>. It is a classical fact, known since the 1950s, that <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> is uncountable for <span><math><mi>c</mi><mo>&gt;</mo><mn>3</mn></math></span> and countable for <span><math><mi>c</mi><mo>&lt;</mo><mn>3</mn></math></span>. However, the cardinality of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> does not appear to be present in the literature. We prove that <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is uncountable.</div><div>More generally, we show that for any <span><math><mi>n</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span>, the set of <span><math><mi>x</mi><mo>∈</mo><mi>R</mi><mo>﹨</mo><mi>Q</mi></math></span> with Lagrange value exactly 3 and such that <span><math><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac><mo>|</mo></mrow><mo>&lt;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span> has exactly <em>n</em> rational solutions <span><math><mfrac><mrow><mi>p</mi></mrow><mrow><mi>q</mi></mrow></mfrac></math></span> is also uncountable.</div></div><div><h3>Video</h3><div>For a video summary of this paper, please visit <span><span>https://youtu.be/VyKB99-kVeY</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 147-173"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790388","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":"p-adic properties of Eisenstein-Kronecker cocycles over imaginary quadratic fields and p-adic interpolation","authors":"Jorge Flórez","doi":"10.1016/j.jnt.2025.11.004","DOIUrl":"10.1016/j.jnt.2025.11.004","url":null,"abstract":"<div><div>We establish integrality and congruence properties for the Eisenstein-Kronecker cocycle introduced by Bergeron, Charollois and García. As a consequence, we recover the integrality of the critical values of Hecke <em>L</em>-functions over imaginary quadratic fields in the split case. Additionally, we construct a <em>p</em>-adic measure that interpolates these critical values.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 48-117"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733458","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":"Hybrid subconvex bounds for GL3 × GL1 twisted L-functions and their applications","authors":"Fei Hou ,&nbsp;GuangShi Lü","doi":"10.1016/j.jnt.2025.09.014","DOIUrl":"10.1016/j.jnt.2025.09.014","url":null,"abstract":"<div><div>Let <span><math><mi>P</mi><mo>,</mo><mi>M</mi></math></span> be two primes such that <span><math><mo>(</mo><mi>P</mi><mo>,</mo><mi>M</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Let <em>f</em> be a Hecke newform of level <em>P</em>, and <em>χ</em> a primitive Dirichlet character modulo <em>M</em>. In this paper, we study the hybrid subconvexity problem for <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><msup><mrow><mtext>sym</mtext></mrow><mrow><mn>2</mn></mrow></msup><mi>f</mi><mo>⊗</mo><mi>χ</mi><mo>)</mo></math></span> simultaneously in the level and conductor aspects. Among other things, we prove that the hybrid subconvex bound can be achieved, so long as <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>&lt;</mo><mi>P</mi><mo>&lt;</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>20</mn></mrow></msup></math></span>. One of the key ingredients is that we develop the classical level aspect version of the Voronoĭ formula for the symmetric square lift in an alternative way by tracing back to its geometric nature. As the direct applications, we obtained the subconvex bound for <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>f</mi><mo>⊗</mo><mi>f</mi><mo>⊗</mo><mi>χ</mi><mo>)</mo></math></span> simultaneously in the level and conductor aspects and the non-obvious bound for the problem of distinguishing modular forms <em>f</em> and <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>χ</mi></mrow></msup></math></span> based on their first Fourier coefficients.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 283-320"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418334","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 local representation densities of Hermitian forms and special cycles II","authors":"Sungyoon Cho","doi":"10.1016/j.jnt.2025.09.020","DOIUrl":"10.1016/j.jnt.2025.09.020","url":null,"abstract":"<div><div>In this paper, we prove that there are certain relations among representation densities and provide an efficient way to compute representation densities by using these relations. As an application, we compute certain arithmetic intersection numbers of special cycles on unitary Rapoport-Zink spaces and propose a conjecture on these.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 169-204"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418998","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":"Scattering constants for some families of noncongruence subgroups","authors":"George Huang ,&nbsp;Henry H. Kim","doi":"10.1016/j.jnt.2025.10.005","DOIUrl":"10.1016/j.jnt.2025.10.005","url":null,"abstract":"<div><div>We compute the scattering constants of three families of noncongruence subgroups, using the Kronecker limit formula.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 533-561"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467141","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":"Asymptotic of the plane overpartition with explicit error terms","authors":"Debika Banerjee ,&nbsp;Arindam Roy","doi":"10.1016/j.jnt.2025.10.010","DOIUrl":"10.1016/j.jnt.2025.10.010","url":null,"abstract":"<div><div>The plane overpartition, a two-dimensional version of the overpartition of an integer <em>n</em>, was introduced recently by Corteel, Savelief, and Vuletić. In the past, this plane overpartition has been studied as the “dotted plane partition” by Brenti, the “strict plane partition” by Vuletić, and the “BKP plane partition” by Foda and Wheeler. In this paper, we establish a strong asymptotic formula for the plane overpartition by giving arbitrarily long summands in the main term and explicit error estimates. In addition, we consider the <em>k</em>-th differences of the plane overpartition and provide a strong asymptotic for these differences. We show that these <em>k</em>-th differences are positive for any fixed <em>k</em> and satisfy higher-order Turán inequalities for any large integer <em>n</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 659-699"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571827","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":"Equidistribution of realizable Steinitz classes for cyclic Kummer extensions","authors":"Brody Lynch","doi":"10.1016/j.jnt.2025.09.022","DOIUrl":"10.1016/j.jnt.2025.09.022","url":null,"abstract":"<div><div>Let <em>ℓ</em> be prime, and <em>K</em> be a number field containing the <em>ℓ</em>-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of <span><math><mi>Z</mi><mo>/</mo><mi>ℓ</mi><mi>Z</mi></math></span> extensions of <em>K</em> ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of <em>K</em>. For <span><math><mi>ℓ</mi><mo>=</mo><mn>2</mn></math></span>, this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-<em>m</em> extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 139-168"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418337","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}