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On Drinfeld modular forms of higher rank VII: Expansions at the boundary
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-11-20 DOI: 10.1016/j.jnt.2024.09.015
Ernst-Ulrich Gekeler
{"title":"On Drinfeld modular forms of higher rank VII: Expansions at the boundary","authors":"Ernst-Ulrich Gekeler","doi":"10.1016/j.jnt.2024.09.015","DOIUrl":"10.1016/j.jnt.2024.09.015","url":null,"abstract":"<div><div>We study expansions of Drinfeld modular forms of rank <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> along the boundary of moduli varieties. Product formulas for the discriminant forms <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are developed, which are analogous with Jacobi's formula for the classical elliptic discriminant. The vanishing orders are described through values at <span><math><mi>s</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>r</mi></math></span> of partial zeta functions of the underlying Drinfeld coefficient ring <em>A</em>. We show linear independence properties for Eisenstein series, which allow to split spaces of modular forms into the subspaces of cusp forms and of Eisenstein series, and give various characterizations of the boundary condition for modular forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 260-340"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An explicit log-free zero density estimate for the Riemann zeta-function
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-11-19 DOI: 10.1016/j.jnt.2024.10.001
Chiara Bellotti
{"title":"An explicit log-free zero density estimate for the Riemann zeta-function","authors":"Chiara Bellotti","doi":"10.1016/j.jnt.2024.10.001","DOIUrl":"10.1016/j.jnt.2024.10.001","url":null,"abstract":"<div><div>We will provide an explicit log-free zero-density estimate for <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></math></span> of the form <span><math><mi>N</mi><mo>(</mo><mi>σ</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>≤</mo><mi>A</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>B</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>σ</mi><mo>)</mo></mrow></msup></math></span>. In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for <span><math><mi>σ</mi><mo>∈</mo><mo>[</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, with <span><math><mn>0.985</mn><mo>≤</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≤</mo><mn>0.9927</mn></math></span> and <span><math><mn>3</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup><mo>&lt;</mo><mi>T</mi><mo>≤</mo><mi>exp</mi><mo>⁡</mo><mo>(</mo><mn>6.7</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 37-77"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A conjecture of Merca on nonnegativity of theta series
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-11-19 DOI: 10.1016/j.jnt.2024.10.003
Bing He, Shuming Liu
{"title":"A conjecture of Merca on nonnegativity of theta series","authors":"Bing He,&nbsp;Shuming Liu","doi":"10.1016/j.jnt.2024.10.003","DOIUrl":"10.1016/j.jnt.2024.10.003","url":null,"abstract":"<div><div>In this paper, we will study a conjecture of Merca on theta series, which gives a refinement of a conjecture of Andrews and Merca on truncated pentagonal number series. We first show refinements of two special cases of Merca's conjecture and then establish several nonnegativity results on theta series. As applications, we establish positivity results involving two celebrated partition statistics.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 17-36"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744195","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 for smooth theta sums with rational parameters
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-11-19 DOI: 10.1016/j.jnt.2024.10.002
Francesco Cellarosi , Tariq Osman
{"title":"Bounds for smooth theta sums with rational parameters","authors":"Francesco Cellarosi ,&nbsp;Tariq Osman","doi":"10.1016/j.jnt.2024.10.002","DOIUrl":"10.1016/j.jnt.2024.10.002","url":null,"abstract":"<div><div>We provide explicit families of pairs <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> such that for sufficiently regular <em>f</em>, there is a constant <em>C</em> for which the theta sum bound<span><span><span><math><mrow><mo>|</mo><munder><mo>∑</mo><mrow><mtext>n</mtext><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></munder><mi>f</mi><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mrow><mi>n</mi></mrow><mo>)</mo></mrow><mi>exp</mi><mo>⁡</mo><mo>{</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo>(</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mo>‖</mo><mrow><mi>n</mi></mrow><mo>‖</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mrow><mi>β</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mi>x</mi><mo>+</mo><mrow><mi>α</mi></mrow><mo>⋅</mo><mrow><mi>n</mi></mrow><mo>)</mo><mo>}</mo><mo>|</mo></mrow><mspace></mspace><mo>≤</mo><mi>C</mi><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> holds for every <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> and every <span><math><mi>N</mi><mo>∈</mo><mi>N</mi></math></span>. Central to the proof is realising that, for fixed <em>N</em>, the theta sum normalised by <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>k</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> agrees with an automorphic function <span><math><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> evaluated along a special curve known as a horocycle lift. The lift depends on the pair <span><math><mo>(</mo><mtext>α</mtext><mo>,</mo><mtext>β</mtext><mo>)</mo></math></span>, and so the bound follows from showing that there are pairs such that <span><math><mo>|</mo><msub><mrow><mi>Θ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>|</mo></math></span> remains bounded along the entire horocycle lift.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 397-426"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744088","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
Accumulation points of normalized approximations 归一化近似值的累积点
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-11-06 DOI: 10.1016/j.jnt.2024.09.002
Kavita Dhanda, Alan Haynes
{"title":"Accumulation points of normalized approximations","authors":"Kavita Dhanda,&nbsp;Alan Haynes","doi":"10.1016/j.jnt.2024.09.002","DOIUrl":"10.1016/j.jnt.2024.09.002","url":null,"abstract":"<div><div>Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points <span><math><mi>q</mi><mi>α</mi></math></span> with <span><math><mi>α</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and <span><math><mi>q</mi><mo>∈</mo><mi>Z</mi></math></span>. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of <strong><em>α</em></strong> whose accumulation points are all of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In the second part we focus primarily on the case when the coordinates of <strong><em>α</em></strong> together with 1 form a basis for an algebraic number field <em>K</em>. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in <em>K</em>) of a single ellipse, or of a pair of hyperbolas, depending on whether or not <em>K</em> has a non-trivial embedding into <span><math><mi>C</mi></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 1-38"},"PeriodicalIF":0.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699240","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 Diophantine equation 2s + pk = m2 with a Fermat prime p 关于 2s + pk = m2 与费马素数 p 的二元一次方程
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-11-05 DOI: 10.1016/j.jnt.2024.09.006
Florian Luca , István Pink
{"title":"On the Diophantine equation 2s + pk = m2 with a Fermat prime p","authors":"Florian Luca ,&nbsp;István Pink","doi":"10.1016/j.jnt.2024.09.006","DOIUrl":"10.1016/j.jnt.2024.09.006","url":null,"abstract":"<div><div>In this paper, we find all the solutions of the Diophantine equation from the title.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 49-71"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699242","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
Statistics for 3-isogeny induced Selmer groups of elliptic curves 椭圆曲线 3-isogeny induced Selmer 群的统计量
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-10-31 DOI: 10.1016/j.jnt.2024.09.003
Pratiksha Shingavekar
{"title":"Statistics for 3-isogeny induced Selmer groups of elliptic curves","authors":"Pratiksha Shingavekar","doi":"10.1016/j.jnt.2024.09.003","DOIUrl":"10.1016/j.jnt.2024.09.003","url":null,"abstract":"<div><div>Given a sixth power free integer <em>a</em>, let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> be the elliptic curve defined by <span><math><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>a</mi></math></span>. We prove explicit results for the lower density of sixth power free integers <em>a</em> for which the 3-isogeny induced Selmer group of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> over <span><math><mi>Q</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> has dimension ≤1. The results are proven by refining the strategy of Davenport–Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for 3-isogeny induced Selmer groups.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 72-94"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699243","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
Construction and non-vanishing of a family of vector-valued Siegel Poincaré series 矢量值西格尔-庞加莱数列族的构造与非消失
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-10-31 DOI: 10.1016/j.jnt.2024.09.007
Sonja Žunar
{"title":"Construction and non-vanishing of a family of vector-valued Siegel Poincaré series","authors":"Sonja Žunar","doi":"10.1016/j.jnt.2024.09.007","DOIUrl":"10.1016/j.jnt.2024.09.007","url":null,"abstract":"<div><div>Using Poincaré series of <em>K</em>-finite matrix coefficients of integrable antiholomorphic discrete series representations of <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, we construct a spanning set for the space <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of Siegel cusp forms of weight <em>ρ</em> for Γ, where <em>ρ</em> is an irreducible polynomial representation of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> of highest weight <span><math><mi>ω</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>…</mo><mo>≥</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>&gt;</mo><mn>2</mn><mi>n</mi></math></span>, and Γ is a discrete subgroup of <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> commensurable with <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 95-123"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699244","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 eventually greedy best underapproximations by Egyptian fractions 关于埃及分数的最终贪婪最佳欠逼近
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-10-31 DOI: 10.1016/j.jnt.2024.09.004
Vjekoslav Kovač
{"title":"On eventually greedy best underapproximations by Egyptian fractions","authors":"Vjekoslav Kovač","doi":"10.1016/j.jnt.2024.09.004","DOIUrl":"10.1016/j.jnt.2024.09.004","url":null,"abstract":"<div><div>Erdős and Graham found it conceivable that the best <em>n</em>-term Egyptian underapproximation of almost every positive number for sufficiently large <em>n</em> gets constructed in a greedy manner, i.e., from the best <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 39-48"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699241","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
Improved bounds for the index conjecture in zero-sum theory 零和理论中指数猜想的改进边界
IF 0.6 3区 数学
Journal of Number Theory Pub Date : 2024-10-31 DOI: 10.1016/j.jnt.2024.09.005
Andrew Pendleton
{"title":"Improved bounds for the index conjecture in zero-sum theory","authors":"Andrew Pendleton","doi":"10.1016/j.jnt.2024.09.005","DOIUrl":"10.1016/j.jnt.2024.09.005","url":null,"abstract":"<div><div>The Index Conjecture in zero-sum theory states that when <em>n</em> is coprime to 6 and <em>k</em> equals 4, every minimal zero-sum sequence of length <em>k</em> modulo <em>n</em> has index 1. While other values of <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> have been studied thoroughly in the last 30 years, it is only recently that the conjecture has been proven for <span><math><mi>n</mi><mo>&gt;</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>20</mn></mrow></msup></math></span>. In this paper, we prove that said upper bound can be reduced to <span><math><mn>4.6</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>13</mn></mrow></msup></math></span>, and lower under certain coprimality conditions. Further, we verify the conjecture for <span><math><mi>n</mi><mo>&lt;</mo><mn>1.8</mn><mo>⋅</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup></math></span> through the application of High Performance Computing (HPC).</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 124-141"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699246","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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