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Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2) GL(2)自同构表示对Hecke特征值的比较
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-10-02 DOI: 10.1016/j.jnt.2025.09.012
Kin Ming Tsang
{"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>&gt;</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}
引用次数: 0
Realisability of simultaneous density constraints for sets of integers 整数集同时密度约束的可实现性
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-10-31 DOI: 10.1016/j.jnt.2025.10.007
Pierre-Yves Bienvenu
{"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}
引用次数: 0
Elliptic curves having non-trivial p-part of Shafarevich-Tate groups and satisfying the Birch and Swinnerton-Dyer conjecture modulo p 具有shafarevic - tate群非平凡p部分且满足Birch和Swinnerton-Dyer猜想模p的椭圆曲线
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-11-19 DOI: 10.1016/j.jnt.2025.10.009
Dongho Byeon, Donggeon Yhee
{"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,&nbsp;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}
引用次数: 0
The Piltz divisor problem in number fields using the resonance method 用共振方法求解数域中的Piltz除数问题
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-11-19 DOI: 10.1016/j.jnt.2025.10.013
Nilmoni Karak, Kamalakshya Mahatab
{"title":"The Piltz divisor problem in number fields using the resonance method","authors":"Nilmoni Karak,&nbsp;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}
引用次数: 0
Analytic rank growth of elliptic curves over cyclic extensions 椭圆曲线在循环扩展上的解析秩增长
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-10-21 DOI: 10.1016/j.jnt.2025.09.026
Gyeongwon Oh , Peter J. Cho
{"title":"Analytic rank growth of elliptic curves over cyclic extensions","authors":"Gyeongwon Oh ,&nbsp;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}
引用次数: 0
Bounds on bilinear sums of generalized Kloosterman sums over arbitrary sets 广义Kloosterman和在任意集合上的双线性和的界
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-10-24 DOI: 10.1016/j.jnt.2025.09.027
Sen Xu , Tianping Zhang
{"title":"Bounds on bilinear sums of generalized Kloosterman sums over arbitrary sets","authors":"Sen Xu ,&nbsp;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>&gt;</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}
引用次数: 0
Greenberg's conjecture and Iwasawa module of real biquadratic fields I 实双二次域的Greenberg猜想与Iwasawa模1
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-10-22 DOI: 10.1016/j.jnt.2025.09.015
Mohamed Mahmoud Chems-Eddin
{"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}
引用次数: 0
Corrigendum to “Invariant part of class groups and distribution of relative class group” [J. Number Theory 279 (2026) 691–748] “类群的不变部分和相对类群的分布”的勘误[J]。数论279 (2026)691-748]
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-11-10 DOI: 10.1016/j.jnt.2025.10.001
Weitong Wang
{"title":"Corrigendum to “Invariant part of class groups and distribution of relative class group” [J. Number Theory 279 (2026) 691–748]","authors":"Weitong Wang","doi":"10.1016/j.jnt.2025.10.001","DOIUrl":"10.1016/j.jnt.2025.10.001","url":null,"abstract":"<div><div>The proof of Lemma 7.10 is flawed. In this corrigendum, we prove the statement under additional conditions which do not affect the statistical results for <span><math><mo>(</mo><mn>6</mn><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span>-fields and <span><math><mo>(</mo><mn>3</mn><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-fields in the introduction part.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 588-595"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520454","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}
引用次数: 0
Improvements on exponential sums related to Piatetski-Shapiro primes 关于Piatetski-Shapiro素数的指数和的改进
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-11-19 DOI: 10.1016/j.jnt.2025.10.015
Li Lu, Lingyu Guo, Victor Zhenyu Guo
{"title":"Improvements on exponential sums related to Piatetski-Shapiro primes","authors":"Li Lu,&nbsp;Lingyu Guo,&nbsp;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}
引用次数: 0
An explicit bound for Siegel zeros and the torsion of elliptic curves with complex multiplication 具有复乘法的椭圆曲线的西格尔零和扭转的显式界
IF 0.7 3区 数学
Journal of Number Theory Pub Date : 2026-04-01 Epub Date: 2025-11-20 DOI: 10.1016/j.jnt.2025.10.014
D. Ralaivaosaona, F.B. Razakarinoro
{"title":"An explicit bound for Siegel zeros and the torsion of elliptic curves with complex multiplication","authors":"D. Ralaivaosaona,&nbsp;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}
引用次数: 0
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