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Proofs of four conjectures of Ballantine, Feigon and Merca on linear inequalities of partitions with odd parts balantine、Feigon和Merca关于奇部分区线性不等式的四个猜想的证明
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-14 DOI: 10.1016/j.jnt.2025.03.014
Olivia X.M. Yao
{"title":"Proofs of four conjectures of Ballantine, Feigon and Merca on linear inequalities of partitions with odd parts","authors":"Olivia X.M. Yao","doi":"10.1016/j.jnt.2025.03.014","DOIUrl":"10.1016/j.jnt.2025.03.014","url":null,"abstract":"<div><div>In their seminal work, Andrews and Merca studied the truncated version of Euler's pentagonal number theorem and deduced an infinite family of linear inequalities for ordinary partition function. The work of Andrews and Merca opened up the study of truncated theta series and linear inequalities for certain restricted partition functions and many articles followed. Recently, Ballantine and Feigon, and Merca posed four conjectures on linear inequalities for partitions with odd parts. In this paper, we confirm those conjectures based on a classical result contributed to Pólya and Szegő.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 344-368"},"PeriodicalIF":0.6,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134733","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
Hecke relations for eta multipliers and congruences for higher-order smallest parts functions 乘子的Hecke关系和高阶最小部函数的同余
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-13 DOI: 10.1016/j.jnt.2025.03.006
Clayton Williams
{"title":"Hecke relations for eta multipliers and congruences for higher-order smallest parts functions","authors":"Clayton Williams","doi":"10.1016/j.jnt.2025.03.006","DOIUrl":"10.1016/j.jnt.2025.03.006","url":null,"abstract":"<div><div>We derive identities from Hecke operators acting on a family of Eisenstein-eta quotients, giving explicit equalities relating the coefficients of these quotients. From these equalities we derive congruences for the coefficients of these Eisenstein-eta quotients modulo powers of primes. As an application we derive systematic congruences for several higher-order smallest parts functions modulo prime powers, resolving a question of Garvan for these cases. We also relate moments of cranks and ranks to the partition function modulo prime powers. Some of our results strengthen and generalize those of a 2023 paper by Wang and Yang.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 325-343"},"PeriodicalIF":0.6,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116414","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
Upper bounds for the lowest first zero in families of cuspidal newforms 尖形新形科中最低首零的上界
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-09 DOI: 10.1016/j.jnt.2025.02.012
Palak Arora , Glenn Bruda , Bruce Fang , Raul Marquez , Steven J. Miller , Beni Prapashtica , Vismay Sharan , Daeyoung Son , Xueyiming Tang , Saad Waheed
{"title":"Upper bounds for the lowest first zero in families of cuspidal newforms","authors":"Palak Arora ,&nbsp;Glenn Bruda ,&nbsp;Bruce Fang ,&nbsp;Raul Marquez ,&nbsp;Steven J. Miller ,&nbsp;Beni Prapashtica ,&nbsp;Vismay Sharan ,&nbsp;Daeyoung Son ,&nbsp;Xueyiming Tang ,&nbsp;Saad Waheed","doi":"10.1016/j.jnt.2025.02.012","DOIUrl":"10.1016/j.jnt.2025.02.012","url":null,"abstract":"<div><div>Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of <em>L</em>-functions lie on the critical line with the real part 1/2. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the <em>n</em>-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within 0.218503 of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 262-289"},"PeriodicalIF":0.6,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069562","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
Distribution of cycles in supersingular ℓ-isogeny graphs 超奇异等构图中环的分布
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-07 DOI: 10.1016/j.jnt.2025.03.013
Eli Orvis
{"title":"Distribution of cycles in supersingular ℓ-isogeny graphs","authors":"Eli Orvis","doi":"10.1016/j.jnt.2025.03.013","DOIUrl":"10.1016/j.jnt.2025.03.013","url":null,"abstract":"<div><div>Recent work by Arpin et al. (2024) <span><span>[2]</span></span> counted the number of cycles of length <em>r</em> in supersingular <em>ℓ</em>-isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine. We provide formulas for both the number of such cycles, and the average number as <span><math><mi>p</mi><mo>→</mo><mo>∞</mo></math></span>, with <em>ℓ</em> and <em>r</em> fixed. In particular, we show that when <em>r</em> is not a power of 2, cycles of length <em>r</em> are disproportionately likely to occur along the spine. We provide experimental evidence that this result holds in the case that <em>r</em> is a power of 2 as well.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 236-261"},"PeriodicalIF":0.6,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144069561","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
Hook length biases for self-conjugate partitions and partitions with distinct odd parts 自共轭分区和奇部明显分区的钩长偏差
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-06 DOI: 10.1016/j.jnt.2025.02.002
Catherine H. Cossaboom
{"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}
引用次数: 0
Tetragonal modular quotients X0+d(N) 四边形模商X0+d(N)
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-06 DOI: 10.1016/j.jnt.2025.03.005
Petar Orlić
{"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}
引用次数: 0
Effectivity for existence of rational points is undecidable 有理点存在的有效性是不可确定的
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-06 DOI: 10.1016/j.jnt.2025.01.023
Natalia Garcia-Fritz , Hector Pasten , Xavier Vidaux
{"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}
引用次数: 0
Generation of cyclotomic Hecke fields by L-values of cusp forms on GL(2) with certain Zp twist 具有一定Zp扭转的GL(2)上尖形的l值生成切环Hecke场
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-06 DOI: 10.1016/j.jnt.2025.02.006
Jaesung Kwon
{"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}
引用次数: 0
Isolations of the sum of two squares from its proper subforms 两个平方和与其固有子形式的分离
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-06 DOI: 10.1016/j.jnt.2025.02.005
Jangwon Ju , Daejun Kim , Kyoungmin Kim , Mingyu Kim , Byeong-Kweon Oh
{"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}
引用次数: 0
Integral points of Shimura varieties: An “all or nothing” principle 志村变异的积分点：一个“全有或全无”的原则
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2025-05-06 DOI: 10.1016/j.jnt.2025.02.003
Haohao Liu
{"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}
引用次数: 0
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