Journal of Number Theory最新文献

{"title":"Rational deformations of the set of multiple zeta-star values","authors":"Jiangtao Li","doi":"10.1016/j.jnt.2025.03.004","DOIUrl":"10.1016/j.jnt.2025.03.004","url":null,"abstract":"<div><div>In this paper we study the derived sets for the rational deformations of multiple zeta-star values. By using the theory of bounded variation functions, we will give function decompositions which describe the metric structure of the derived sets. The connection between the rational deformation of multiple zeta-star values and the <em>n</em>-th Cantor set in fractal geometry is also discussed.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 23-56"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917289","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 small value set polynomials over finite fields and monodromy groups","authors":"Xiantao Deng,&nbsp;Bin Xu,&nbsp;Qianxi Zhu","doi":"10.1016/j.jnt.2025.03.008","DOIUrl":"10.1016/j.jnt.2025.03.008","url":null,"abstract":"<div><div>In this article, we study polynomials over finite fields with small value sets through their monodromy groups. More specifically, we consider polynomials <span><math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span> such that <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo></math></span> is Galois, which we call absolutely minimal value set polynomials (AMVSPs) over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We examine their relationship with minimal value set polynomials (MVSPs) by directly utilizing tools and results from function field theory. In particular, we prove that if <span><math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span> satisfies <span><math><mn>1</mn><mo>&lt;</mo><mi>deg</mi><mo>⁡</mo><mo>(</mo><mi>f</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></math></span>, then <span><math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is an MVSP if and only if it is an AMVSP. As an application of our results, we provide a simple proof of the classification results of Mills (see <span><span>[16, Theorem 2]</span></span>) and Borges-Conceição (see <span><span>[4, Theorem 2.3]</span></span>) for MVSPs with degree <span><math><mo>≤</mo><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></math></span> using a completely different approach.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 139-161"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935259","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":"Divisibility of the multiplicative order modulo monic irreducible polynomials over finite fields","authors":"Joaquim Cera Da Conceição","doi":"10.1016/j.jnt.2025.03.003","DOIUrl":"10.1016/j.jnt.2025.03.003","url":null,"abstract":"<div><div>We consider the set of monic irreducible polynomials <em>P</em> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> such that the multiplicative order modulo <em>P</em> of some <em>a</em> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is divisible by a fixed positive integer <em>d</em>. Call <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> this set. We show the existence or non-existence of the density of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> for three distinct notions of density. In particular, the sets <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> have a Dirichlet density. Under some assumptions, we prove simple formulas for the density values.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 105-123"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942531","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":"Multiplicative generalised polynomial sequences","authors":"Jakub Konieczny","doi":"10.1016/j.jnt.2025.03.007","DOIUrl":"10.1016/j.jnt.2025.03.007","url":null,"abstract":"<div><div>We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the sequence is not zero almost everywhere.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 147-164"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942533","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":"Maesaka–Seki–Watanabe's formula for multiple harmonic q-sums","authors":"Yuto Tsuruta","doi":"10.1016/j.jnt.2025.02.009","DOIUrl":"10.1016/j.jnt.2025.02.009","url":null,"abstract":"<div><div>Maesaka, Seki, and Watanabe recently discovered an equality called the MSW formula. This paper provides a <em>q</em>-analogue of MSW formula. It discusses a new proof of the duality relation for finite multiple harmonic <em>q</em>-series at primitive roots of unity via <em>q</em>-analogue of MSW formula. This paper also gives a <em>q</em>-analogue of Yamamoto's generalization of MSW formula for Schur type.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 270-285"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943228","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":"Cyclotomic primes","authors":"Carl Pomerance","doi":"10.1016/j.jnt.2025.02.013","DOIUrl":"10.1016/j.jnt.2025.02.013","url":null,"abstract":"<div><div>Mersenne primes and Fermat primes may be thought of as primes of the form <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the <em>m</em>th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 198-208"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935256","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":"Determination of quadratic lattices by local structure and sublattices of codimension one","authors":"N.D. Meyer ,&nbsp;A.G. Earnest","doi":"10.1016/j.jnt.2025.03.011","DOIUrl":"10.1016/j.jnt.2025.03.011","url":null,"abstract":"<div><div>For definite quadratic lattices over the rings of integers of totally real algebraic number fields, it is shown that lattices are determined up to isometry by their local structure and sublattices of codimension one. In particular, a theorem of Kitaoka for <span><math><mi>Z</mi></math></span>-lattices is generalized to definite lattices over totally real algebraic number fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 232-256"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935349","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":"Ibukiyama correspondences on automorphic forms on Mp4(AQ) and SO5(AQ) generating large discrete series representations at the real place","authors":"Hiroshi Ishimoto","doi":"10.1016/j.jnt.2025.03.002","DOIUrl":"10.1016/j.jnt.2025.03.002","url":null,"abstract":"<div><div>In our previous paper we gave proofs of Ibukiyama's correspondences on holomorphic Siegel modular forms of degree 2 of half-integral weight and integral weight. In this paper, we formulate and prove similar correspondences on automorphic forms on <span><math><msub><mrow><mi>Mp</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>)</mo></math></span> or <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> generating large discrete series representations at the real components. In addition, we show that the correspondences can be described in terms of local theta correspondences.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 63-104"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942029","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":"Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms","authors":"Chengliang Guo","doi":"10.1016/j.jnt.2025.03.009","DOIUrl":"10.1016/j.jnt.2025.03.009","url":null,"abstract":"<div><div>Let <em>ψ</em> be a smooth compactly supported function on <span><math><mi>X</mi><mo>=</mo><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>H</mi></math></span>. In this paper, we are interested in the joint cubic moments of automorphic forms when the spectral parameters go to infinity. We show that the diagonal case for Eisenstein series <span><math><msub><mrow><mo>∫</mo></mrow><mrow><mi>X</mi></mrow></msub><mi>ψ</mi><mo>(</mo><mi>z</mi><mo>)</mo><mi>E</mi><msup><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>i</mi><mi>t</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mi>d</mi><mi>μ</mi><mi>z</mi><mo>=</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>ψ</mi></mrow></msub><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>. In off-diagonal case we prove <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mi>t</mi></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>X</mi></mrow></msub><mi>ψ</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>z</mi><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>i</mi><mi>t</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>g</mi><mo>(</mo><mi>z</mi><mo>)</mo><mi>d</mi><mi>μ</mi><mi>z</mi><mo>=</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> as long as <span><math><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>t</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>}</mo><mo>→</mo><mo>∞</mo></math></span>. Finally we show <span><math><msub><mrow><mo>∫</mo></mrow><mrow><mi>X</mi></mrow></msub><mi>ψ</mi><mo>(</mo><mi>z</mi><mo>)</mo><msup><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>z</mi><mo>)</mo><mi>g</mi><mo>(</mo><mi>z</mi><mo>)</mo><mi>d</mi><mi>μ</mi><mi>z</mi><mo>=</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> in the range <span><math><mo>|</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>−</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>|</mo><mo>≤</mo><msubsup><mrow><mi>t</mi></mrow><mrow><mi>f</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn><mo>−</mo><mi>ε</mi></mrow></msubsup></math></span> where <span><math><mi>f</mi><mo>,</mo><mi>g</mi></math></span> are two Hecke-Maass cusp forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 162-197"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935255","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":"On the signed Selmer groups for motives at non-ordinary primes in Zp2-extensions","authors":"Jishnu Ray ,&nbsp;Florian Ito Sprung","doi":"10.1016/j.jnt.2025.02.011","DOIUrl":"10.1016/j.jnt.2025.02.011","url":null,"abstract":"<div><div>In this paper, we give certain arithmetic applications of the multi-signed Coleman maps constructed by the first author and Cédric Dion over the <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>-extension of an imaginary quadratic field for non-ordinary motives. Our first result is a control theorem for multi-signed Selmer groups over the <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>-extension and our second result is the variation of the Iwasawa <em>μ</em>-invariants for two such representations which are congruent modulo <em>p</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 209-231"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935348","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}