Pub Date : 2025-12-03DOI: 10.1016/j.jnt.2025.11.005
Enrique González–Jiménez , Nguyen Xuan Tho
Let D be a square-free integer. Under certain conditions on D, we characterize non-constant arithmetic progressions of squares over quadratic extensions of .
设D是一个无平方整数。在D上的一定条件下，刻画了Q(D)的二次扩展上的非常等差数列。
{"title":"Squares in arithmetic progression over certain non-primitive quartic number fields","authors":"Enrique González–Jiménez , 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":"2025-12-03","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}
Pub Date : 2025-12-03DOI: 10.1016/j.jnt.2025.11.004
Jorge Flórez
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 L-functions over imaginary quadratic fields in the split case. Additionally, we construct a p-adic measure that interpolates these critical values.
我们建立了Bergeron， Charollois和García引入的Eisenstein-Kronecker循环的完整性和同余性质。因此，我们恢复了分裂情况下虚二次域上Hecke l -函数的临界值的完整性。此外，我们构造了一个p进测度来插值这些临界值。
{"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":"2025-12-03","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}
Pub Date : 2025-11-20DOI: 10.1016/j.jnt.2025.10.012
Alessandro Languasco , Rashi Lunia , Pieter Moree
Already Dedekind and Weber considered the problem of counting integral ideals of norm at most x in a given number field K. Here we improve on the existing results in case 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 x.
{"title":"Counting ideals in abelian number fields","authors":"Alessandro Languasco , Rashi Lunia , 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":"2025-11-20","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}
Pub Date : 2025-11-20DOI: 10.1016/j.jnt.2025.10.014
D. Ralaivaosaona, F.B. Razakarinoro
For any integer such that −d is a fundamental discriminant, we show that the Dirichlet L-function associated with the real primitive character does not vanish on the positive part of the interval . As an application of this result, we prove that the size of the torsion subgroup of an elliptic curve with complex multiplication over a degree d number field is bounded above by for .
对于任意整数d≥3且−d是一个基本判判式，我们证明了与实基元特征χ(⋅)=(−d⋅)相关的Dirichlet l -函数在区间[1−6.035/d，1]的正部不消失。作为这一结果的一个应用，我们证明了在d次数域上具有复数乘法的椭圆曲线的扭转子群的大小在d≥3⋅108时有390dlog log log d的上界。
{"title":"An explicit bound for Siegel zeros and the torsion of elliptic curves with complex multiplication","authors":"D. Ralaivaosaona, F.B. Razakarinoro","doi":"10.1016/j.jnt.2025.10.014","DOIUrl":"10.1016/j.jnt.2025.10.014","url":null,"abstract":"<div><div>For any integer <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> such that −<em>d</em> is a fundamental discriminant, we show that the Dirichlet <em>L</em>-function associated with the real primitive character <span><math><mi>χ</mi><mo>(</mo><mo>⋅</mo><mo>)</mo><mo>=</mo><mo>(</mo><mfrac><mrow><mo>−</mo><mi>d</mi></mrow><mrow><mo>⋅</mo></mrow></mfrac><mo>)</mo></math></span> does not vanish on the positive part of the interval <span><math><mo>[</mo><mn>1</mn><mo>−</mo><mn>6.035</mn><mo>/</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>,</mo><mspace></mspace><mn>1</mn><mo>]</mo></math></span>. As an application of this result, we prove that the size of the torsion subgroup of an elliptic curve with complex multiplication over a degree <em>d</em> number field is bounded above by <span><math><mn>390</mn><mi>d</mi><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>d</mi></math></span> for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>8</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 795-829"},"PeriodicalIF":0.7,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618276","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.010
Debika Banerjee , Arindam Roy
The plane overpartition, a two-dimensional version of the overpartition of an integer n, 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 k-th differences of the plane overpartition and provide a strong asymptotic for these differences. We show that these k-th differences are positive for any fixed k and satisfy higher-order Turán inequalities for any large integer n.
{"title":"Asymptotic of the plane overpartition with explicit error terms","authors":"Debika Banerjee , 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":"2025-11-19","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.009
Dongho Byeon, Donggeon Yhee
In this paper, we prove that for a family of elliptic curves defined over , there are infinitely many quadratic twists having non-trivial p-part of Shafarevich-Tate groups and satisfying a weak form of the Birch and Swinnerton-Dyer conjecture modulo p, where .
{"title":"Elliptic curves having non-trivial p-part of Shafarevich-Tate groups and satisfying the Birch and Swinnerton-Dyer conjecture modulo p","authors":"Dongho Byeon, Donggeon Yhee","doi":"10.1016/j.jnt.2025.10.009","DOIUrl":"10.1016/j.jnt.2025.10.009","url":null,"abstract":"<div><div>In this paper, we prove that for a family of elliptic curves defined over <span><math><mi>Q</mi></math></span>, there are infinitely many quadratic twists having non-trivial <em>p</em>-part of Shafarevich-Tate groups and satisfying a weak form of the Birch and Swinnerton-Dyer conjecture modulo <em>p</em>, where <span><math><mi>p</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 648-658"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571769","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.013
Nilmoni Karak, Kamalakshya Mahatab
The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved Ω-bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.
{"title":"The Piltz divisor problem in number fields using the resonance method","authors":"Nilmoni Karak, Kamalakshya Mahatab","doi":"10.1016/j.jnt.2025.10.013","DOIUrl":"10.1016/j.jnt.2025.10.013","url":null,"abstract":"<div><div>The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved Ω-bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 726-740"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618277","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.015
Li Lu, Lingyu Guo, Victor Zhenyu Guo
We prove a new bound to the exponential sum of the form by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form . In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in a Beatty sequence and so on.
我们用I型和的一种新方法证明了形式为∑h ~ h δh∑m ~ m∑n ~ Nmn ~ xambne（αmn+h(mn+u)γ）的指数和的一个新界。这个和可以应用于许多与皮亚茨基-夏皮罗素数有关的问题，它是形式为⌊nc⌋的素数。本文改进了Balog-Friedlander条件的可容许范围，从而改进了带Piatetski-Shapiro素数的三元哥德巴赫问题。我们还研究了等差数列中的Piatetski-Shapiro素数的分布，Beatty数列中的Piatetski-Shapiro素数等。
{"title":"Improvements on exponential sums related to Piatetski-Shapiro primes","authors":"Li Lu, Lingyu Guo, Victor Zhenyu Guo","doi":"10.1016/j.jnt.2025.10.015","DOIUrl":"10.1016/j.jnt.2025.10.015","url":null,"abstract":"<div><div>We prove a new bound to the exponential sum of the form<span><span><span><math><munder><mo>∑</mo><mrow><mi>h</mi><mo>∼</mo><mi>H</mi></mrow></munder><msub><mrow><mi>δ</mi></mrow><mrow><mi>h</mi></mrow></msub><munder><mrow><munder><mo>∑</mo><mrow><mi>m</mi><mo>∼</mo><mi>M</mi></mrow></munder><munder><mo>∑</mo><mrow><mi>n</mi><mo>∼</mo><mi>N</mi></mrow></munder></mrow><mrow><mi>m</mi><mi>n</mi><mo>∼</mo><mi>x</mi></mrow></munder><mspace></mspace><msub><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msub><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>e</mi><mo>(</mo><mi>α</mi><mi>m</mi><mi>n</mi><mo>+</mo><mi>h</mi><msup><mrow><mo>(</mo><mi>m</mi><mi>n</mi><mo>+</mo><mi>u</mi><mo>)</mo></mrow><mrow><mi>γ</mi></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span> by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form <span><math><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>⌋</mo></math></span>. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in a Beatty sequence and so on.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 700-725"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618248","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.011
Muhammad Afifurrahman
We obtain an asymptotic formula for the number of integer matrices that have determinant Δ and whose absolute values of the entries are at most H. The result holds uniformly for a large range of Δ with respect to H.
{"title":"A uniform formula on the number of integer matrices with given determinant and height","authors":"Muhammad Afifurrahman","doi":"10.1016/j.jnt.2025.10.011","DOIUrl":"10.1016/j.jnt.2025.10.011","url":null,"abstract":"<div><div>We obtain an asymptotic formula for the number of integer <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices that have determinant Δ and whose absolute values of the entries are at most <em>H</em>. The result holds uniformly for a large range of Δ with respect to <em>H</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 741-770"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617773","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}
Pub Date : 2025-11-19DOI: 10.1016/j.jnt.2025.10.016
Peter Vang Uttenthal
In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on that remain inequivalent modulo 3 for and 20887, and that these ℓ are the three smallest such primes.
{"title":"Parameters of solvable automorphic forms","authors":"Peter Vang Uttenthal","doi":"10.1016/j.jnt.2025.10.016","DOIUrl":"10.1016/j.jnt.2025.10.016","url":null,"abstract":"<div><div>In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>)</mo></math></span> that remain inequivalent modulo 3 for <span><math><mi>ℓ</mi><mo>=</mo><mn>7687</mn><mo>,</mo><mn>16363</mn></math></span> and 20887, and that these <em>ℓ</em> are the three smallest such primes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 771-794"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618275","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}
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