Pub Date : 2026-04-01Epub Date: 2025-10-02DOI: 10.1016/j.jnt.2025.09.012
Kin Ming Tsang
We consider a variant of the strong multiplicity one theorem. Let and be two unitary cuspidal automorphic representations for that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which , where is the trace of the Langlands conjugacy class of at v. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which .
{"title":"Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)","authors":"Kin Ming Tsang","doi":"10.1016/j.jnt.2025.09.012","DOIUrl":"10.1016/j.jnt.2025.09.012","url":null,"abstract":"<div><div>We consider a variant of the strong multiplicity one theorem. Let <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> be two unitary cuspidal automorphic representations for <span><math><mi>GL</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span> that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which <span><math><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>|</mo></mrow><mo>></mo><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>|</mo></mrow></math></span>, where <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is the trace of the Langlands conjugacy class of <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> at <em>v</em>. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which <span><math><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>|</mo></mrow><mo>≠</mo><mrow><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>|</mo></mrow></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 1-42"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326834","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 : 2026-04-01Epub Date: 2025-10-31DOI: 10.1016/j.jnt.2025.10.007
Pierre-Yves Bienvenu
In this note, we study the set of values of the quadruplet where and denote the lower and upper asymptotic density, respectively. Complementing existing results on the topic, we determine each of its six projections on coordinate planes, that is, the sets of possible values of the six subpairs of the quadruplet. Further, we show that this set has non empty interior, in particular has positive measure. To do so, we use among others probabilistic and diophantine methods. Some auxiliary results pertaining to these methods may be of general interest.
在这篇文章中，我们研究了四重态(d_(A), D (A),d_（2A), D (2A)）的值的集合D，其中A∧N和d_， D分别表示下和上渐近密度。补充已有的结果，我们确定了它在坐标平面上的六个投影，即四重体的六个子对的可能值的集合。进一步证明了该集合D具有非空的内部，特别是具有正测度。为此，我们使用概率和丢番图方法。与这些方法有关的一些辅助结果可能会引起一般的兴趣。
{"title":"Realisability of simultaneous density constraints for sets of integers","authors":"Pierre-Yves Bienvenu","doi":"10.1016/j.jnt.2025.10.007","DOIUrl":"10.1016/j.jnt.2025.10.007","url":null,"abstract":"<div><div>In this note, we study the set <span><math><mi>D</mi></math></span> of values of the quadruplet <span><math><mo>(</mo><munder><mrow><mi>d</mi></mrow><mo>_</mo></munder><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><munder><mrow><mi>d</mi></mrow><mo>_</mo></munder><mo>(</mo><mn>2</mn><mi>A</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mo>‾</mo></mover><mo>(</mo><mn>2</mn><mi>A</mi><mo>)</mo><mo>)</mo></math></span> where <span><math><mi>A</mi><mo>⊂</mo><mi>N</mi></math></span> and <span><math><munder><mrow><mi>d</mi></mrow><mo>_</mo></munder><mo>,</mo><mover><mrow><mi>d</mi></mrow><mo>‾</mo></mover></math></span> denote the lower and upper asymptotic density, respectively. Complementing existing results on the topic, we determine each of its six projections on coordinate planes, that is, the sets of possible values of the six subpairs of the quadruplet. Further, we show that this set <span><math><mi>D</mi></math></span> has non empty interior, in particular has positive measure. To do so, we use among others probabilistic and diophantine methods. Some auxiliary results pertaining to these methods may be of general interest.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 596-614"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520452","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub Date: 2025-10-21DOI: 10.1016/j.jnt.2025.09.026
Gyeongwon Oh , Peter J. Cho
Let E be an elliptic curve defined over . For an odd prime l, we consider the family of degree l cyclic extensions K over . When we view the elliptic curve E as a curve over K, the analytic rank of the L-function of E over K may increase compared to that of the L-function of E over . Under the generalized Riemann hypothesis, we demonstrate the rarity of significant increases in analytic ranks.
{"title":"Analytic rank growth of elliptic curves over cyclic extensions","authors":"Gyeongwon Oh , Peter J. Cho","doi":"10.1016/j.jnt.2025.09.026","DOIUrl":"10.1016/j.jnt.2025.09.026","url":null,"abstract":"<div><div>Let <em>E</em> be an elliptic curve defined over <span><math><mi>Q</mi></math></span>. For an odd prime <em>l</em>, we consider the family of degree <em>l</em> cyclic extensions <em>K</em> over <span><math><mi>Q</mi></math></span>. When we view the elliptic curve <em>E</em> as a curve over <em>K</em>, the analytic rank of the <em>L</em>-function <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> of <em>E</em> over <em>K</em> may increase compared to that of the <em>L</em>-function <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> of <em>E</em> over <span><math><mi>Q</mi></math></span>. Under the generalized Riemann hypothesis, we demonstrate the rarity of significant increases in analytic ranks.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 267-282"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419013","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 : 2026-04-01Epub Date: 2025-10-24DOI: 10.1016/j.jnt.2025.09.027
Sen Xu , Tianping Zhang
Text
We prove a new bound on bilinear forms with generalized Kloosterman sums by using the sum-product phenomenon in , which reached the barrier for the more general situation and complements those obtained by Kowalski, Michel, and Sawin (2020). We also establish new estimates for bilinear forms with two variables from arbitrary subsets of , which has expanded the range of obtained by Xi (2023).
Video
For a video summary of this paper, please visit https://youtu.be/Q472zpufLEs.
本文利用Fp中的和积现象证明了广义Kloosterman和双线性形式的一个新界，该界在更一般的情况下达到了MN>；p34，并补充了Kowalski， Michel, and Sawin（2020）的结果。我们还从Fp的任意子集中建立了具有两个变量的双线性形式的新估计，这扩展了Xi（2023）得到的M，N的范围。观看本文的视频摘要，请访问https://youtu.be/Q472zpufLEs。
{"title":"Bounds on bilinear sums of generalized Kloosterman sums over arbitrary sets","authors":"Sen Xu , Tianping Zhang","doi":"10.1016/j.jnt.2025.09.027","DOIUrl":"10.1016/j.jnt.2025.09.027","url":null,"abstract":"<div><h3>Text</h3><div>We prove a new bound on bilinear forms with generalized Kloosterman sums by using the sum-product phenomenon in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, which reached the barrier <span><math><mi>M</mi><mi>N</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></math></span> for the more general situation and complements those obtained by Kowalski, Michel, and Sawin (2020). We also establish new estimates for bilinear forms with two variables from arbitrary subsets of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, which has expanded the range of <span><math><mi>M</mi><mo>,</mo><mi>N</mi></math></span> obtained by Xi (2023).</div></div><div><h3>Video</h3><div>For a video summary of this paper, please visit <span><span>https://youtu.be/Q472zpufLEs</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 373-396"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418338","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 : 2026-04-01Epub Date: 2025-10-22DOI: 10.1016/j.jnt.2025.09.015
Mohamed Mahmoud Chems-Eddin
The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem:
What are real biquadratic number fields k such that ? where is the 2-Iwasawa module of k and is the 2-class group of the first layer of the cyclotomic -extension of k. Moreover, we give several families of real biquadratic fields k such that is trivial or isomorphic to or , where n is a given positive integer. The reader can also find some results concerning the 2-rank of the class group of certain real triquadratic fields.
{"title":"Greenberg's conjecture and Iwasawa module of real biquadratic fields I","authors":"Mohamed Mahmoud Chems-Eddin","doi":"10.1016/j.jnt.2025.09.015","DOIUrl":"10.1016/j.jnt.2025.09.015","url":null,"abstract":"<div><div>The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem:</div><div>What are real biquadratic number fields <em>k</em> such that <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo><mo>)</mo><mo>=</mo><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo></math></span>? where <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span> is the 2-Iwasawa module of <em>k</em> and <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the 2-class group of <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> the first layer of the cyclotomic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-extension of <em>k</em>. Moreover, we give several families of real biquadratic fields <em>k</em> such that <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span> is trivial or isomorphic to <span><math><mi>Z</mi><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>Z</mi></math></span> or <span><math><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>Z</mi></math></span>, where <em>n</em> is a given positive integer. The reader can also find some results concerning the 2-rank of the class group of certain real triquadratic fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 224-266"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418339","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 : 2026-04-01Epub Date: 2025-11-03DOI: 10.1016/j.jnt.2025.10.004
Tim Gehrunger, Richard Pink
Consider a hyperelliptic curve of genus g over a field K of characteristic zero. After extending K we can view it as a marked curve with its Weierstrass points. We present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic 2 over a finite extension of K. In the cases we work out relatively simple conditions for the structure of this reduction.
{"title":"Reduction of hyperelliptic curves in residue characteristic 2","authors":"Tim Gehrunger, Richard Pink","doi":"10.1016/j.jnt.2025.10.004","DOIUrl":"10.1016/j.jnt.2025.10.004","url":null,"abstract":"<div><div>Consider a hyperelliptic curve of genus <em>g</em> over a field <em>K</em> of characteristic zero. After extending <em>K</em> we can view it as a marked curve with its <span><math><mn>2</mn><mi>g</mi><mo>+</mo><mn>2</mn></math></span> Weierstrass points. We present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic 2 over a finite extension of <em>K</em>. In the cases <span><math><mi>g</mi><mo>⩽</mo><mn>2</mn></math></span> we work out relatively simple conditions for the structure of this reduction.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 429-491"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467142","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub 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":"2026-04-01","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号