Pub Date : 2026-02-06DOI: 10.1016/j.jnt.2026.01.008
Sebastián Herrero , Tobías Martínez , Pedro Montero
Inspired by Bourqui's work on anticanonical height zeta functions on Hirzebruch surfaces, we study height zeta functions of complete smooth split toric varieties with Picard rank 2 over global function fields, with respect to height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where precise analytic properties of the corresponding height zeta functions can be given. As application, we obtain asymptotic formulas for the number of rational points of large height on each subvariety, with explicit leading constants and controlled error terms.
{"title":"Counting rational points on Hirzebruch–Kleinschmidt varieties over global function fields","authors":"Sebastián Herrero , Tobías Martínez , Pedro Montero","doi":"10.1016/j.jnt.2026.01.008","DOIUrl":"10.1016/j.jnt.2026.01.008","url":null,"abstract":"<div><div>Inspired by Bourqui's work on anticanonical height zeta functions on Hirzebruch surfaces, we study height zeta functions of complete smooth split toric varieties with Picard rank 2 over global function fields, with respect to height functions associated with big metrized line bundles. We show that these varieties can be naturally decomposed into a finite disjoint union of subvarieties, where precise analytic properties of the corresponding height zeta functions can be given. As application, we obtain asymptotic formulas for the number of rational points of large height on each subvariety, with explicit leading constants and controlled error terms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 1-53"},"PeriodicalIF":0.7,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147485","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}
Pub Date : 2026-02-06DOI: 10.1016/j.jnt.2025.12.015
Iván Blanco-Chacón , Luis Dieulefait
Let be a totally real number field and an ideal of its ring of integers of norm N. Let be a prime totally split in F such that . For every even , define the -dimensional parallel weight . Let be any non CM Hilbert cuspidal Hecke eigenform. Assume that the residual representation has large image for some prime over p in the field of definition of f. Under these conditions, we prove that there exists a lift of associated to a Hilbert modular cuspform which is supercuspidal at each prime of F over p. We also give a proof of the corresponding statement for classical Hecke cuspforms. Such statement was already proved by Khare [23] with classical techniques. Finally, using our main result we give a corrigenda for [12], correctly inserting the micro good dihedral prime in the level.
{"title":"Modular supercuspidal lifts of weight 2","authors":"Iván Blanco-Chacón , Luis Dieulefait","doi":"10.1016/j.jnt.2025.12.015","DOIUrl":"10.1016/j.jnt.2025.12.015","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi><mo>/</mo><mi>Q</mi></math></span> be a totally real number field and <span><math><mi>N</mi></math></span> an ideal of its ring of integers of norm <em>N</em>. Let <span><math><mi>p</mi><mo>></mo><mi>max</mi><mo></mo><mo>{</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>}</mo></math></span> be a prime totally split in <em>F</em> such that <span><math><mi>p</mi><mo>∤</mo><mi>N</mi></math></span>. For every even <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, define the <span><math><mo>[</mo><mi>F</mi><mo>:</mo><mi>Q</mi><mo>]</mo></math></span>-dimensional parallel weight <span><math><mtext>k</mtext><mo>=</mo><mo>(</mo><mi>k</mi><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>k</mi><mo>)</mo></math></span>. Let <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext>k</mtext></mrow></msub><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>)</mo></math></span> be any non CM Hilbert cuspidal Hecke eigenform. Assume that the residual representation <span><math><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>f</mi><mo>,</mo><mi>P</mi></mrow></msub></math></span> has large image for some prime <span><math><mi>P</mi></math></span> over <em>p</em> in the field of definition of <em>f</em>. Under these conditions, we prove that there exists a lift of <span><math><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>f</mi><mo>,</mo><mi>P</mi></mrow></msub></math></span> associated to a Hilbert modular cuspform <span><math><mi>g</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext>2</mtext></mrow></msub><mo>(</mo><mi>N</mi><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>ϵ</mi><mo>)</mo></math></span> which is supercuspidal at each prime of <em>F</em> over <em>p</em>. We also give a proof of the corresponding statement for classical Hecke cuspforms. Such statement was already proved by Khare <span><span>[23]</span></span> with classical techniques. Finally, using our main result we give a corrigenda for <span><span>[12]</span></span>, correctly inserting the micro good dihedral prime in the level.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"285 ","pages":"Pages 54-73"},"PeriodicalIF":0.7,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147486","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}
Let G be a finite abelian group and k be an integer not dividing the exponent of G. We denote by the smallest positive integer l such that every sequence over G of length no less than l has a zero-sum subsequence of length not divisible by k. In this paper, we focus on determining for , a cyclic group of order n. Specifically, we prove that for .
{"title":"On the existence of zero-sum subsequences with length not divided by a given number","authors":"Weidong Gao , Xiao Jiang , Yuanlin Li , Huijuan Qi","doi":"10.1016/j.jnt.2025.12.002","DOIUrl":"10.1016/j.jnt.2025.12.002","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group and <em>k</em> be an integer not dividing the exponent of <em>G</em>. We denote by <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> the smallest positive integer <em>l</em> such that every sequence over <em>G</em> of length no less than <em>l</em> has a zero-sum subsequence of length not divisible by <em>k</em>. In this paper, we focus on determining <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, a cyclic group of order <em>n</em>. Specifically, we prove that<span><span><span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>⌊</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⌋</mo><mo>+</mo><mn>1</mn></math></span></span></span> for <span><math><mi>k</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>}</mo><mo>∪</mo><mo>(</mo><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 15-37"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080811","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}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.003
Ethan Simpson Lee , Paweł Nosal
In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that and are bounded from above by for all and respectively, where and are the Chebyshev ψ and ϑ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.
{"title":"Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis","authors":"Ethan Simpson Lee , Paweł Nosal","doi":"10.1016/j.jnt.2025.12.003","DOIUrl":"10.1016/j.jnt.2025.12.003","url":null,"abstract":"<div><div>In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that <span><math><mo>|</mo><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>x</mi><mo>|</mo></math></span> and <span><math><mo>|</mo><mi>ϑ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>x</mi><mo>|</mo></math></span> are bounded from above by<span><span><span><math><mfrac><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt><mi>log</mi><mo></mo><mi>x</mi><mo>(</mo><mi>log</mi><mo></mo><mi>x</mi><mo>−</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>8</mn><mi>π</mi></mrow></mfrac></math></span></span></span> for all <span><math><mi>x</mi><mo>≥</mo><mn>101</mn></math></span> and <span><math><mi>x</mi><mo>≥</mo><mn>2</mn><mspace></mspace><mn>657</mn></math></span> respectively, where <span><math><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>ϑ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are the Chebyshev <em>ψ</em> and <em>ϑ</em> functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 241-258"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023475","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}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.001
Seokhyun Choi, Bo-Hae Im
We prove Larsen's conjecture for elliptic curves over with analytic rank at most 1. Specifically, let be an elliptic curve over . If has analytic rank at most 1, then we prove that for any topologically finitely generated subgroup G of , the rank of E over the fixed subfield of under G is infinite.
{"title":"Larsen's conjecture for elliptic curves over Q with analytic rank at most one","authors":"Seokhyun Choi, Bo-Hae Im","doi":"10.1016/j.jnt.2025.12.001","DOIUrl":"10.1016/j.jnt.2025.12.001","url":null,"abstract":"<div><div>We prove Larsen's conjecture for elliptic curves over <span><math><mi>Q</mi></math></span> with analytic rank at most 1. Specifically, let <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> be an elliptic curve over <span><math><mi>Q</mi></math></span>. If <span><math><mi>E</mi><mo>/</mo><mi>Q</mi></math></span> has analytic rank at most 1, then we prove that for any topologically finitely generated subgroup <em>G</em> of <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover><mo>/</mo><mi>Q</mi><mo>)</mo></math></span>, the rank of <em>E</em> over the fixed subfield <span><math><msup><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msup></math></span> of <span><math><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></math></span> under <em>G</em> is infinite.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"284 ","pages":"Pages 1-14"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039934","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}
Pub Date : 2026-01-20DOI: 10.1016/j.jnt.2025.12.006
Biao Wang , Shaoyun Yi
Let be an irreducible polynomial of degree . Let be an integer. The number of integers n such that is k-free is widely studied in the literature. In principle, one expects that is k-free infinitely often, if f has no fixed k-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.
{"title":"The prime number theorem over integers of power-free polynomial values","authors":"Biao Wang , Shaoyun Yi","doi":"10.1016/j.jnt.2025.12.006","DOIUrl":"10.1016/j.jnt.2025.12.006","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> be an irreducible polynomial of degree <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span>. Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer. The number of integers <em>n</em> such that <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is <em>k</em>-free is widely studied in the literature. In principle, one expects that <span><math><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is <em>k</em>-free infinitely often, if <em>f</em> has no fixed <em>k</em>-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 216-229"},"PeriodicalIF":0.7,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023473","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}
Pub Date : 2026-01-19DOI: 10.1016/j.jnt.2025.12.007
Kegang Liu , Zicheng Qian
We prove a mod-p local-global compatibility result for Scholze's functor in higher dimensions, under certain multiplicity-free condition. This improves the previous result in this direction of K. Liu, by removing the semisimple assumption on the mod p Galois representations. Our proof relies mainly on a criterion for σ-typicity of modules which is obtained by representation-theoretic techniques.
{"title":"A note on mod-p local-global compatibility via Scholze's functor","authors":"Kegang Liu , Zicheng Qian","doi":"10.1016/j.jnt.2025.12.007","DOIUrl":"10.1016/j.jnt.2025.12.007","url":null,"abstract":"<div><div>We prove a mod-<em>p</em> local-global compatibility result for Scholze's functor in higher dimensions, under certain multiplicity-free condition. This improves the previous result in this direction of K. Liu, by removing the semisimple assumption on the mod <em>p</em> Galois representations. Our proof relies mainly on a criterion for <em>σ</em>-typicity of modules which is obtained by representation-theoretic techniques.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 230-240"},"PeriodicalIF":0.7,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023474","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}
Pub Date : 2025-12-30DOI: 10.1016/j.jnt.2025.11.007
Pedro-José Cazorla García
Given a prime number q and a squarefree integer , we develop a method to explicitly determine the tuples for which the difference has squarefree part equal to . Our techniques include the combination of the local information provided by Galois representations of Frey–Hellegouarch curves with the effective resolution of Thue–Mahler equations, as well as the use of improved lower bounds for q-adic and complex logarithms. As an application of this methodology, we will completely resolve the case when and .
{"title":"On differences of perfect powers and prime powers","authors":"Pedro-José Cazorla García","doi":"10.1016/j.jnt.2025.11.007","DOIUrl":"10.1016/j.jnt.2025.11.007","url":null,"abstract":"<div><div>Given a prime number <em>q</em> and a squarefree integer <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, we develop a method to explicitly determine the tuples <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>α</mi><mo>)</mo></math></span> for which the difference <span><math><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span> has squarefree part equal to <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. Our techniques include the combination of the local information provided by Galois representations of Frey–Hellegouarch curves with the effective resolution of Thue–Mahler equations, as well as the use of improved lower bounds for <em>q</em>-adic and complex logarithms. As an application of this methodology, we will completely resolve the case when <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mn>20</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>q</mi><mo><</mo><mn>25</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 121-169"},"PeriodicalIF":0.7,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927960","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}
Pub Date : 2025-12-29DOI: 10.1016/j.jnt.2025.11.011
Naveen K. Godara , Renu Joshi , Eshita Mazumdar
This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order , where p is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite 2-groups.
{"title":"Combinatorial invariants for certain classes of non-abelian groups","authors":"Naveen K. Godara , Renu Joshi , Eshita Mazumdar","doi":"10.1016/j.jnt.2025.11.011","DOIUrl":"10.1016/j.jnt.2025.11.011","url":null,"abstract":"<div><div>This article focuses on the study of zero-sum invariants of finite non-abelian groups. We address two main problems: the first centers on the ordered Davenport constant and the second on Gao's constant. We establish a connection between the ordered Davenport constant and the small Davenport constant for a finite non-abelian group of even order, which in turn gives a relation with the Noether number. Additionally, we confirm a conjecture of Gao and Li for a non-abelian group of order <span><math><mn>2</mn><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>, where <em>p</em> is a prime. Furthermore, we prove a conjecture that connects the ordered Davenport constant to the Loewy length for certain classes of finite 2-groups.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 44-63"},"PeriodicalIF":0.7,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886043","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}
Pub Date : 2025-12-29DOI: 10.1016/j.jnt.2025.11.010
Yeong-Wook Kwon , Subong Lim
In this paper, we investigate a Zagier duality between the Fourier coefficients of harmonic Maass–Jacobi–Poincaré series and those of weakly skew-holomorphic Jacobi–Poincaré series. We also verify a similar duality involving the skew-holomorphic Jacobi–Eisenstein series. As an application of these duality results, we show that the weakly skew-holomorphic Poincaré series and the skew-holomorphic Jacobi–Eisenstein series are orthogonal to the space of skew-holomorphic Jacobi cusp forms. Moreover, in the case of integral weight and level one, we obtain the rationality for the coefficients of the skew-holomorphic Jacobi–Eisenstein series. Combined with the duality result for the Jacobi–Eisenstein series, this implies the rationality of the constant term in the holomorphic part of the harmonic Maass–Jacobi–Poincaré series.
{"title":"Zagier duality for Jacobi forms","authors":"Yeong-Wook Kwon , Subong Lim","doi":"10.1016/j.jnt.2025.11.010","DOIUrl":"10.1016/j.jnt.2025.11.010","url":null,"abstract":"<div><div>In this paper, we investigate a Zagier duality between the Fourier coefficients of harmonic Maass–Jacobi–Poincaré series and those of weakly skew-holomorphic Jacobi–Poincaré series. We also verify a similar duality involving the skew-holomorphic Jacobi–Eisenstein series. As an application of these duality results, we show that the weakly skew-holomorphic Poincaré series and the skew-holomorphic Jacobi–Eisenstein series are orthogonal to the space of skew-holomorphic Jacobi cusp forms. Moreover, in the case of integral weight and level one, we obtain the rationality for the coefficients of the skew-holomorphic Jacobi–Eisenstein series. Combined with the duality result for the Jacobi–Eisenstein series, this implies the rationality of the constant term in the holomorphic part of the harmonic Maass–Jacobi–Poincaré series.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 15-43"},"PeriodicalIF":0.7,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886044","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}
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