Journal of Number Theory最新文献

{"title":"Infinitude of the zeros of the Lerch zeta function on the half plane ℜ(s)>1","authors":"Biswajyoti Saha ,&nbsp;Dhananjaya Sahu","doi":"10.1016/j.jnt.2025.08.019","DOIUrl":"10.1016/j.jnt.2025.08.019","url":null,"abstract":"<div><div>For <span><math><mi>a</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, the zeros of the Hurwitz zeta function <span><math><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup></math></span> have interesting features. There are no zeros in the half plane <span><math><mo>ℜ</mo><mo>(</mo><mi>s</mi><mo>)</mo><mo>≥</mo><mn>1</mn><mo>+</mo><mi>a</mi></math></span>, whereas there are infinitely many zeros in the strip <span><math><mn>1</mn><mo>&lt;</mo><mo>ℜ</mo><mo>(</mo><mi>s</mi><mo>)</mo><mo>&lt;</mo><mn>1</mn><mo>+</mo><mi>a</mi></math></span>, provided <span><math><mi>a</mi><mo>≠</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>1</mn></math></span>. The existence of these infinitely many zeros was first proved by Davenport and Heilbronn for rational and transcendental values of <em>a</em> and then by Cassels for algebraic irrational values of <em>a</em>. In this article, we consider the analogous question for the zeros of the cognate Lerch zeta function <span><math><msub><mrow><mi>ζ</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo>)</mo><mo>:</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup></math></span>, where <em>z</em> is a complex number of unit modulus. When <em>z</em> is a root of unity, the question can be answered using a theorem of Zaghloul, which is an extension of a work of Chatterjee and Gun. In the general case, we need to further extend the method of Chatterjee and Gun.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 506-518"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220656","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":"An extension of smooth numbers: Multiple dense divisibility","authors":"Garo Sarajian ,&nbsp;Andreas Weingartner","doi":"10.1016/j.jnt.2025.08.013","DOIUrl":"10.1016/j.jnt.2025.08.013","url":null,"abstract":"<div><div>The <em>i</em>-tuply <em>y</em>-densely divisible numbers were introduced by a Polymath project, as a weaker condition on the moduli than <em>y</em>-smoothness, in distribution estimates for primes in arithmetic progressions. We obtain the order of magnitude of the count of these integers up to <em>x</em>, uniformly in <em>x</em> and <em>y</em>, for every fixed natural number <em>i</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 278-317"},"PeriodicalIF":0.7,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158520","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":"Torsion of rational elliptic curves over the Zp-extensions of quadratic fields","authors":"Ömer Avcı","doi":"10.1016/j.jnt.2025.08.009","DOIUrl":"10.1016/j.jnt.2025.08.009","url":null,"abstract":"<div><div>Let <em>E</em> be an elliptic curve defined over <span><math><mi>Q</mi></math></span>. For a quadratic number field <em>K</em> and an odd prime number <em>p</em>, let <em>L</em> be a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension of <em>K</em>. We prove that <span><math><mi>E</mi><msub><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow><mrow><mtext>tors</mtext></mrow></msub><mo>=</mo><mi>E</mi><msub><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mrow><mtext>tors</mtext></mrow></msub></math></span> when <span><math><mi>p</mi><mo>&gt;</mo><mn>5</mn></math></span>. It enables us to classify the groups that can be realized as the torsion subgroup <span><math><mi>E</mi><msub><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow><mrow><mtext>tors</mtext></mrow></msub></math></span>, by using the classification of torsion subgroups over the quadratic fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 153-170"},"PeriodicalIF":0.7,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096122","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":"Galois theory of quadratic rational functions with periodic critical points","authors":"Özlem Ejder","doi":"10.1016/j.jnt.2025.08.010","DOIUrl":"10.1016/j.jnt.2025.08.010","url":null,"abstract":"<div><div>Given a number field <em>k</em>, and a quadratic rational function <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, the associated arboreal representation of the absolute Galois group of <em>k</em> is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for <span><math><mi>f</mi><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the <em>n</em>th level of the tree remains small as <em>n</em> goes to infinity.</div><div>In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of <em>f</em>, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs.</div><div>Furthermore, we study the iterated extension <span><math><mi>k</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span> generated by adjoining to <span><math><mi>k</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> all roots of <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>t</mi></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> for a parameter <em>t</em>. We call the intersection of <span><math><mi>k</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span> with <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span>, the field of constants associated with <em>f</em>. When one of the two critical points of <em>f</em> is the image of the other, we show that the field of constants is contained in the cyclotomic extension of <em>k</em> generated by all 2-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 212-245"},"PeriodicalIF":0.7,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096120","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":"Abelian varieties with real multiplication: Classification and isogeny classes over finite fields","authors":"Tejasi Bhatnagar ,&nbsp;Yu Fu","doi":"10.1016/j.jnt.2025.08.006","DOIUrl":"10.1016/j.jnt.2025.08.006","url":null,"abstract":"<div><div>In this paper, we provide a classification of certain points on Hilbert modular varieties over finite fields under a mild assumption on Newton polygon. Furthermore, we use this characterization to prove estimates for the size of isogeny classes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 171-190"},"PeriodicalIF":0.7,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096123","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":"Non-vanishing of a certain quantity related to the p-adic coupling of mock modular forms with newforms","authors":"Pavel Guerzhoy","doi":"10.1016/j.jnt.2025.08.007","DOIUrl":"10.1016/j.jnt.2025.08.007","url":null,"abstract":"<div><div>Several authors have recently proved results which express a cusp form as a <em>p</em>-adic limit of weakly holomorphic modular forms under repeated application of Atkin's <em>U</em>-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart <span><span>[1]</span></span> found a method to prove the non-vanishing in question in the specific case considered by El-Guindy and Ono <span><span>[10]</span></span>. Hanson and Jameson <span><span>[15]</span></span> and (independently) Dicks <span><span>[8]</span></span> generalized this method to finitely many other cases.</div><div>In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono <span><span>[10]</span></span>, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's <em>p</em>-adic Γ-function.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 191-211"},"PeriodicalIF":0.7,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096119","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":"Monodromy of elliptic logarithms: Some topological methods and effective results","authors":"Francesco Tropeano","doi":"10.1016/j.jnt.2025.08.008","DOIUrl":"10.1016/j.jnt.2025.08.008","url":null,"abstract":"<div><div>We study monodromy groups associated with elliptic schemes, examining the action induced by the fundamental group of the base via analytic continuation. We develop effective methods for investigating the relative monodromy group of elliptic logarithms and present explicit constructions of loops that simultaneously have trivial action on periods and non-trivial action on logarithms. We provide a new proof that the relative monodromy group of non-torsion sections has full rank. Our results include topological methods and effective techniques for analyzing the ramification locus of sections.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 49-87"},"PeriodicalIF":0.7,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049554","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 moments of averages of Ramanujan sums","authors":"Shivani Goel ,&nbsp;M. Ram Murty","doi":"10.1016/j.jnt.2025.08.002","DOIUrl":"10.1016/j.jnt.2025.08.002","url":null,"abstract":"<div><div>Chan and Kumchev studied averages of the first and second moments of Ramanujan sums. In this article, we extend this investigation by estimating the higher moments of averages of Ramanujan sums using a Tauberian theorem due to La Bretèche. We also give a result for the moments of averages of Cohen-Ramanujan sums.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 987-1003"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018763","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":"Corresponding Abelian extensions of integrally equivalent number fields","authors":"Shaver Phagan","doi":"10.1016/j.jnt.2025.08.001","DOIUrl":"10.1016/j.jnt.2025.08.001","url":null,"abstract":"<div><div>Extensive work has been done to determine necessary and sufficient conditions for a bijective correspondence of abelian extensions of number fields to force an isomorphism of the base fields. However, explicit examples of correspondences over non-isomorphic fields are rare. Integrally equivalent number fields admit an induced correspondence of abelian extensions. Studying this correspondence using idelic class field theory and linear algebra, we show that the corresponding extensions share features similar to those of arithmetically equivalent fields, and yet they are not generally weakly Kronecker equivalent. We also extend a group cohomological result of Arapura-Katz-McReynolds-Solapurkar and present geometric and arithmetic applications.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 88-112"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060280","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":"Bounds on the number of squares in recurrence sequences: y0 = b2 (I)","authors":"Paul M. Voutier","doi":"10.1016/j.jnt.2025.08.003","DOIUrl":"10.1016/j.jnt.2025.08.003","url":null,"abstract":"<div><div>We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mo>∞</mo></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span>, arising from the solutions of generalised negative Pell equations, <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>d</mi><msup><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>c</mi></math></span>, where −<em>c</em> and <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are any positive squares. We show that there are at most 2 distinct squares larger than an explicit lower bound in such sequences. From this result, we also show that there are at most 5 distinct squares when <span><math><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for infinitely many values of <em>b</em>, including all <span><math><mn>1</mn><mo>≤</mo><mi>b</mi><mo>≤</mo><mn>24</mn></math></span>, as well as once <em>d</em> exceeds an explicit lower bound, without any conditions on the size of such squares.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 246-270"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096118","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}