Journal of Number Theory最新文献

{"title":"Hook length biases for self-conjugate partitions and partitions with distinct odd parts","authors":"Catherine H. Cossaboom","doi":"10.1016/j.jnt.2025.02.002","DOIUrl":"10.1016/j.jnt.2025.02.002","url":null,"abstract":"<div><div>We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span> among self-conjugate partitions of <em>n</em> than among partitions of distinct odd parts of <em>n</em> for sufficiently large <em>n</em>. More precisely, we derive asymptotic formulas for the total number of hooks of fixed length <em>t</em> in both classes. This resolves a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 290-324"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099347","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":"Tetragonal modular quotients X0+d(N)","authors":"Petar Orlić","doi":"10.1016/j.jnt.2025.03.005","DOIUrl":"10.1016/j.jnt.2025.03.005","url":null,"abstract":"<div><div>Let <em>N</em> be a positive integer. For every <span><math><mi>d</mi><mo>|</mo><mi>N</mi></math></span> such that <span><math><mo>(</mo><mi>d</mi><mo>,</mo><mi>N</mi><mo>/</mo><mi>d</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> there exists an Atkin-Lehner involution <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> of the modular curve <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>. In this paper we determine all quotient curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> whose <span><math><mi>Q</mi></math></span>-gonality is equal to 4 and all quotient curves <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> whose <span><math><mi>C</mi></math></span>-gonality is equal to 4.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 98-114"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935258","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":"Effectivity for existence of rational points is undecidable","authors":"Natalia Garcia-Fritz ,&nbsp;Hector Pasten ,&nbsp;Xavier Vidaux","doi":"10.1016/j.jnt.2025.01.023","DOIUrl":"10.1016/j.jnt.2025.01.023","url":null,"abstract":"<div><div>The analogue of Hilbert's tenth problem over <span><math><mi>Q</mi></math></span> asks for an algorithm to decide the existence of rational points on algebraic varieties over this field. This remains as one of the main open problems in the area of undecidability in number theory. Besides the existence of rational points, there is also considerable interest in the problem of effectivity: one asks whether the sought rational points satisfy determined height bounds, often expressed in terms of the height of the coefficients of the equations defining the algebraic varieties under consideration. We show that, in fact, Hilbert's tenth problem over <span><math><mi>Q</mi></math></span> with (finitely many) height comparison conditions is undecidable.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 81-97"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935260","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":"Generation of cyclotomic Hecke fields by L-values of cusp forms on GL(2) with certain Zp twist","authors":"Jaesung Kwon","doi":"10.1016/j.jnt.2025.02.006","DOIUrl":"10.1016/j.jnt.2025.02.006","url":null,"abstract":"<div><div>Let <em>F</em> be a number field, <em>f</em> an algebraic automorphic newform on <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math></span> over <em>F</em>, <em>p</em> an odd prime does not divide the class number of <em>F</em> and the level of <em>f</em>. We prove that <em>f</em> is determined by its <em>L</em>-values twisted by Galois characters <em>ϕ</em> of certain <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension of <em>F</em>. Furthermore, if <em>F</em> is totally real or CM, then under some mild assumption on <em>f</em>, the compositum of the Hecke field of <em>f</em> and the cyclotomic field <span><math><mi>Q</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></math></span> is generated by the algebraic <em>L</em>-values of <em>f</em> twisted by Galois characters <em>ϕ</em> of certain <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension of <em>F</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 115-138"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935261","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":"Isolations of the sum of two squares from its proper subforms","authors":"Jangwon Ju ,&nbsp;Daejun Kim ,&nbsp;Kyoungmin Kim ,&nbsp;Mingyu Kim ,&nbsp;Byeong-Kweon Oh","doi":"10.1016/j.jnt.2025.02.005","DOIUrl":"10.1016/j.jnt.2025.02.005","url":null,"abstract":"<div><div>For a (positive definite and integral) quadratic form <em>f</em>, a quadratic form is said to be <em>an isolation of f from its proper subforms</em> if it represents all proper subforms of <em>f</em>, but not <em>f</em> itself. It was proved that the minimal rank of isolations of the square quadratic form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is three, and there are exactly 15 ternary diagonal isolations of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and there are quinary isolations of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. In this article, we prove that there are at most 231 quinary isolations of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, which are listed in Table 1. Moreover, we prove that 14 quinary quadratic forms with dagger mark in Table 1 are isolations of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 1-18"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143937942","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":"Integral points of Shimura varieties: An “all or nothing” principle","authors":"Haohao Liu","doi":"10.1016/j.jnt.2025.02.003","DOIUrl":"10.1016/j.jnt.2025.02.003","url":null,"abstract":"<div><div>For algebraic varieties over number fields, we define a locus that measures the infiniteness of integral points (of chosen integral models). For a Shimura variety <em>S</em>, Lang's conjecture predicts that the locus of <em>S</em> is empty when the level structure is high, and we prove that this locus is either empty or <em>S</em> itself.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 124-146"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942532","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":"The irrationality of an infinite series involving ω(n) under a prime tuples conjecture","authors":"Kyle Pratt","doi":"10.1016/j.jnt.2025.02.010","DOIUrl":"10.1016/j.jnt.2025.02.010","url":null,"abstract":"<div><div>Let <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of distinct prime factors of <em>n</em>. Assuming a suitably uniform version of the prime <em>k</em>-tuples conjecture, we show that the number<span><span><span><math><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mfrac><mrow><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac></mrow></math></span></span></span> is irrational. This settles (conditionally) a question of Erdős.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 57-71"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917290","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":"Class numbers, congruent numbers and umbral moonshine","authors":"Miranda C.N. Cheng ,&nbsp;John F.R. Duncan ,&nbsp;Michael H. Mertens","doi":"10.1016/j.jnt.2025.02.007","DOIUrl":"10.1016/j.jnt.2025.02.007","url":null,"abstract":"<div><div>In earlier work we initiated a program to study relationships between finite groups and arithmetic geometric invariants of modular curves in a systematic way. In the present work we continue this program, with a focus on the two smallest sporadic simple Mathieu groups. To do this we first elucidate a connection between a special case of umbral moonshine and the imaginary quadratic class numbers. Then we use this connection to classify a distinguished set of modules for the smallest sporadic Mathieu group. Finally we establish a connection between our classification and the congruent number problem of antiquity.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 201-235"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069560","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":"Restricted free products of Demushkin groups of rank ℵ0 as absolute Galois groups","authors":"Tamar Bar-On","doi":"10.1016/j.jnt.2025.03.012","DOIUrl":"10.1016/j.jnt.2025.03.012","url":null,"abstract":"<div><div>We prove that a restricted free profinite (pro-<em>p</em>) product over a countable set of pro-<em>p</em> Demushkin groups of rank <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, that can be realized as absolute Galois groups, is isomorphic to an absolute Galois group if and only if <span><math><msub><mrow><mi>log</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⁡</mo><mi>q</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 257-269"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935257","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":"Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture","authors":"Victor Zhenyu Guo,&nbsp;Yuan Yi","doi":"10.1016/j.jnt.2025.02.008","DOIUrl":"10.1016/j.jnt.2025.02.008","url":null,"abstract":"<div><div>The Piatetski-Shapiro sequences are of the form <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></msup><mo>:</mo><mo>=</mo><msubsup><mrow><mo>(</mo><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>⌋</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> with <span><math><mi>c</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>∉</mo><mi>N</mi></math></span>. Let <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the sequence of primes in ascending order. In this paper, we study the distribution of pairs <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> of consecutive primes such that <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></msup></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></msup></math></span> for <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≠</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and give a conjecture with the prime counting functions of the pairs <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>. We give a heuristic argument to support this prediction based on a model by Lemke Oliver and Soundararajan which relies on a strong form of the Hardy-Littlewood conjecture. Moreover, we prove a proposition related to the average of singular series with a weight of a complex exponential function.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 286-314"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943230","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}