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Bounds for the relative class number problem for function fields 函数域的相对类数问题的边界
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-19 DOI: 10.1016/j.jnt.2025.05.010
Santiago Arango-Piñeros , María Chara , Asimina S. Hamakiotes , Kiran S. Kedlaya , Gustavo Rama
{"title":"Bounds for the relative class number problem for function fields","authors":"Santiago Arango-Piñeros ,&nbsp;María Chara ,&nbsp;Asimina S. Hamakiotes ,&nbsp;Kiran S. Kedlaya ,&nbsp;Gustavo Rama","doi":"10.1016/j.jnt.2025.05.010","DOIUrl":"10.1016/j.jnt.2025.05.010","url":null,"abstract":"<div><div>We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 977-1010"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522824","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
Conic bundles and Mordell–Weil ranks of elliptic surfaces 椭圆曲面的圆锥束和莫德尔-韦尔列
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-19 DOI: 10.1016/j.jnt.2025.05.013
Felipe Zingali Meira
{"title":"Conic bundles and Mordell–Weil ranks of elliptic surfaces","authors":"Felipe Zingali Meira","doi":"10.1016/j.jnt.2025.05.013","DOIUrl":"10.1016/j.jnt.2025.05.013","url":null,"abstract":"<div><div>Let <em>k</em> be a number field and <span><math><mi>E</mi></math></span> an elliptic curve defined over the function field <span><math><mi>k</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> given by an equation of the form <span><math><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>k</mi><mo>[</mo><mi>T</mi><mo>]</mo></math></span> and <span><math><mi>deg</mi><mo>⁡</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mn>2</mn></math></span>. We explore the conic bundle structure over the <em>x</em>-line to obtain lower and upper bounds for the Mordell–Weil rank of <span><math><mi>E</mi><mo>(</mo><mi>k</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 786-815"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491589","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
Inducing contractions of the mother of all continued fractions 所有连分式的母体的诱导收缩
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-19 DOI: 10.1016/j.jnt.2025.05.015
Karma Dajani , Cor Kraaikamp , Slade Sanderson
{"title":"Inducing contractions of the mother of all continued fractions","authors":"Karma Dajani ,&nbsp;Cor Kraaikamp ,&nbsp;Slade Sanderson","doi":"10.1016/j.jnt.2025.05.015","DOIUrl":"10.1016/j.jnt.2025.05.015","url":null,"abstract":"<div><div>We introduce a new, large class of continued fraction algorithms producing what are called <em>contracted Farey expansions</em>. These algorithms are defined by coupling two acceleration techniques—<em>induced transformations</em> and <em>contraction</em>—in the setting of Shunji Ito's (<span><span>[19]</span></span>) natural extension of the Farey tent map, which generates ‘slow’ continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's <em>S</em>-expansions, and Nakada's parameterised family of <em>α</em>-continued fractions for all <span><math><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>≤</mo><mn>1</mn></math></span> as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the <em>α</em>-continued fraction transformations as an explicit induced transformation of Ito's natural extension.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 816-874"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491592","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
A Kronecker congruence relation for modular functions of higher level and genus 高阶和属模函数的Kronecker同余关系
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-19 DOI: 10.1016/j.jnt.2025.05.011
Ho Yun Jung , Ja Kyung Koo , Dong Hwa Shin
{"title":"A Kronecker congruence relation for modular functions of higher level and genus","authors":"Ho Yun Jung ,&nbsp;Ja Kyung Koo ,&nbsp;Dong Hwa Shin","doi":"10.1016/j.jnt.2025.05.011","DOIUrl":"10.1016/j.jnt.2025.05.011","url":null,"abstract":"<div><div>Let <em>j</em> be the elliptic modular function, a weakly holomorphic modular function for <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. Weber showed that for each prime <em>p</em> the modular polynomial <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mspace></mspace><mi>y</mi><mo>)</mo></math></span> of <em>j</em> satisfies what is known as the Kronecker congruence relation <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mspace></mspace><mi>y</mi><mo>)</mo><mo>≡</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>−</mo><mi>y</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>−</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo><mspace></mspace><mo>(</mo><mtext>mod</mtext><mspace></mspace><mi>p</mi><mi>Z</mi><mo>[</mo><mi>x</mi><mo>,</mo><mspace></mspace><mi>y</mi><mo>]</mo><mo>)</mo></math></span>. We give a generalization of this congruence applicable to certain weakly holomorphic modular functions of higher level in terms of integrality over <span><math><mi>Z</mi><mo>[</mo><mi>j</mi><mo>]</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 875-892"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491593","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
Congruences modulo arbitrary powers of 5 and 7 for Andrews and Paule's partition diamonds with (n + 1) copies of n 对（n + 1）个n拷贝的Andrews和Paule分割菱形取5和7的任意幂模的同余
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-19 DOI: 10.1016/j.jnt.2025.05.004
Julia Q.D. Du , Olivia X.M. Yao
{"title":"Congruences modulo arbitrary powers of 5 and 7 for Andrews and Paule's partition diamonds with (n + 1) copies of n","authors":"Julia Q.D. Du ,&nbsp;Olivia X.M. Yao","doi":"10.1016/j.jnt.2025.05.004","DOIUrl":"10.1016/j.jnt.2025.05.004","url":null,"abstract":"<div><div>Recently, Andrews and Paule introduced a partition function <span><math><mi>P</mi><mi>D</mi><mi>N</mi><mn>1</mn><mo>(</mo><mi>N</mi><mo>)</mo></math></span> which denotes the number of partition diamonds with <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> copies of <em>n</em> where summing the parts at the links gives <em>N</em>. They also presented the generating function for <span><math><mi>P</mi><mi>D</mi><mi>N</mi><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and proved several congruences modulo 5,7,25,49 for <span><math><mi>P</mi><mi>D</mi><mi>N</mi><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical <span><math><mi>p</mi><mo>(</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>k</mi></mrow></msup><mi>n</mi><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mn>5</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>, where <span><math><mn>24</mn><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mn>5</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> and <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of 5 for <span><math><mi>P</mi><mi>D</mi><mi>N</mi><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. In addition, we prove two congruences modulo arbitrary powers of 7 for <span><math><mi>P</mi><mi>D</mi><mi>N</mi><mn>1</mn><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, which are analogous to Watson's congruences for <span><math><mi>p</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 1-44"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490676","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
Galois scaffolds for extraspecial p-extensions in characteristic 0 特征值为0的超种p扩展的伽罗瓦支架
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-19 DOI: 10.1016/j.jnt.2025.05.005
Kevin Keating , Paul Schwartz
{"title":"Galois scaffolds for extraspecial p-extensions in characteristic 0","authors":"Kevin Keating ,&nbsp;Paul Schwartz","doi":"10.1016/j.jnt.2025.05.005","DOIUrl":"10.1016/j.jnt.2025.05.005","url":null,"abstract":"<div><div>Let <em>K</em> be a local field of characteristic 0 with residue characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>. Let <em>G</em> be an extraspecial <em>p</em>-group and let <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> be a totally ramified <em>G</em>-extension. In this paper we find sufficient conditions for <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span> to be free of rank 1 over its associated order <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>L</mi><mo>/</mo><mi>K</mi></mrow></msub></math></span>, and to stricter conditions which imply that <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>L</mi><mo>/</mo><mi>K</mi></mrow></msub></math></span> is a Hopf order in the group ring <span><math><mi>K</mi><mo>[</mo><mi>G</mi><mo>]</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 893-923"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491594","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
Almost primes between all squares 所有平方之间几乎都是质数
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-18 DOI: 10.1016/j.jnt.2025.05.009
Adrian W. Dudek , Daniel R. Johnston
{"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}
引用次数: 0
An alternative Q-form of the cyclotomic double shuffle Lie algebra 分环双洗牌李代数的另一种q型
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-18 DOI: 10.1016/j.jnt.2025.05.006
Hidekazu Furusho, Khalef Yaddaden
{"title":"An alternative Q-form of the cyclotomic double shuffle Lie algebra","authors":"Hidekazu Furusho,&nbsp;Khalef Yaddaden","doi":"10.1016/j.jnt.2025.05.006","DOIUrl":"10.1016/j.jnt.2025.05.006","url":null,"abstract":"<div><div>We present an alternative <span><math><mi>Q</mi></math></span>-form for Racinet's cyclotomic double shuffle Lie algebra, inspired by the double shuffle relations among congruent multiple zeta values studied by Yuan and Zhao. Our main result establishes an invariance characterization theorem, demonstrating how these two <span><math><mi>Q</mi></math></span>-forms can be reconstructed from each other under Galois action.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 953-976"},"PeriodicalIF":0.6,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490943","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
On the average congruence class bias for cyclicity and divisibility of the groups of Fp-points of elliptic curves 椭圆曲线fp点群的循环性和可整除性的平均同余类偏置
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-18 DOI: 10.1016/j.jnt.2025.05.003
Sung Min Lee
{"title":"On the average congruence class bias for cyclicity and divisibility of the groups of Fp-points of elliptic curves","authors":"Sung Min Lee","doi":"10.1016/j.jnt.2025.05.003","DOIUrl":"10.1016/j.jnt.2025.05.003","url":null,"abstract":"<div><div>In 2009, W.D. Banks and I.E. Shparlinski studied the average densities of primes <span><math><mi>p</mi><mo>≤</mo><mi>x</mi></math></span> for which the reductions of elliptic curves of small height modulo <em>p</em> satisfy certain algebraic properties, namely cyclicity and divisibility of the number of points by a fixed integer <em>m</em>. In this paper, we refine their results by restricting the primes <em>p</em> under consideration to lie in an arithmetic progression <span><math><mi>k</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>n</mi></math></span>. Furthermore, for a fixed modulus <em>n</em>, we investigate statistical biases among the different congruence classes <span><math><mi>k</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>n</mi></math></span> of primes satisfying the aforementioned properties.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 746-785"},"PeriodicalIF":0.6,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491591","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
Fine Selmer groups of CM elliptic curves CM椭圆曲线的精细Selmer群
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-06-18 DOI: 10.1016/j.jnt.2025.05.012
Meng Fai Lim , Jun Wang , Dong Yan
{"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}
引用次数: 0
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