Pub Date : 2025-09-05DOI: 10.1016/j.jnt.2025.08.006
Tejasi Bhatnagar , Yu Fu
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.
{"title":"Abelian varieties with real multiplication: Classification and isogeny classes over finite fields","authors":"Tejasi Bhatnagar , 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}
Pub Date : 2025-09-05DOI: 10.1016/j.jnt.2025.08.007
Pavel Guerzhoy
Several authors have recently proved results which express a cusp form as a p-adic limit of weakly holomorphic modular forms under repeated application of Atkin's U-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 [1] found a method to prove the non-vanishing in question in the specific case considered by El-Guindy and Ono [10]. Hanson and Jameson [15] and (independently) Dicks [8] generalized this method to finitely many other cases.
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 [10], where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's p-adic Γ-function.
{"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}
Pub Date : 2025-09-05DOI: 10.1016/j.jnt.2025.08.008
Francesco Tropeano
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.
{"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}
Pub Date : 2025-09-04DOI: 10.1016/j.jnt.2025.08.002
Shivani Goel , M. Ram Murty
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.
{"title":"On the moments of averages of Ramanujan sums","authors":"Shivani Goel , 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}
Pub Date : 2025-09-04DOI: 10.1016/j.jnt.2025.08.001
Shaver Phagan
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.
{"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}
Pub Date : 2025-09-04DOI: 10.1016/j.jnt.2025.08.003
Paul M. Voutier
We continue and generalise our earlier investigations of the number of squares in binary recurrence sequences. Here we consider sequences, , arising from the solutions of generalised negative Pell equations, , where −c and 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 for infinitely many values of b, including all , as well as once d exceeds an explicit lower bound, without any conditions on the size of such squares.
{"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}
Pub Date : 2025-09-04DOI: 10.1016/j.jnt.2025.08.004
Petr Kucheriaviy
<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo></mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1
如果自然数A中的任何元素都不能整除其他自然数，则称自然数A为本数。设Ω(n)为n具有多重性的质因数个数。设fz(A)=∑A∈AzΩ(A) A (log (A) z，其中z∈R>；0。Erdős在1935年证明了f1(A)=∑A∈A1alog (A)在所有原始集A上是一致有界的。我们证明了Erdős不等式的一个推广，对于z∈(0,2)时的fz(A)提供了一个类似的结果。进一步，我们研究了fz(A)在所有原始集合上的最优性。我们还讨论了limz→0 (A)，它是Dirichlet密度的推广。我们研究了fz（Pk）的渐近性，其中Pk={n：Ω(n)=k}。对于z=1，我们找到f1（Pk）的渐近展开中的下一项，改进了Gorodetsky， Lichtman和Wong的结果。我们还研究了∑a∈AzΩ(a)/a在[1，N]的所有原始子集上的最优性。
{"title":"Erdős inequality for primitive sets","authors":"Petr Kucheriaviy","doi":"10.1016/j.jnt.2025.08.004","DOIUrl":"10.1016/j.jnt.2025.08.004","url":null,"abstract":"<div><div>A set of natural numbers <em>A</em> is called primitive if no element of <em>A</em> divides any other. Let <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of prime divisors of <em>n</em> counted with multiplicity. Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup></mrow><mrow><mi>a</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>z</mi></mrow></msup></mrow></mfrac></math></span>, where <span><math><mi>z</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>. Erdős proved in 1935 that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mi>log</mi><mo></mo><mi>a</mi></mrow></mfrac></math></span> is uniformly bounded over all primitive sets <em>A</em>. We prove a generalization of Erdős inequality which provides an analogous result for <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, when <span><math><mi>z</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. Furthermore, we study the supremum of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> over all primitive sets. We also discuss <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>z</mi><mo>→</mo><mn>0</mn></mrow></msub><mo></mo><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, which is a generalization of Dirichlet density. We study the asymptotics of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>{</mo><mi>n</mi><mo>:</mo><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. For <span><math><mi>z</mi><mo>=</mo><mn>1</mn></math></span> we find the next term in asymptotic expansion of <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> refining the result of Gorodetsky, Lichtman, and Wong. We also study the supremum of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>Ω</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msup><mo>/</mo><mi>a</mi></math></span> over all primitive subsets of <span><math><mo>[</mo><mn>1","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 113-152"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096121","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 : 2025-09-04DOI: 10.1016/j.jnt.2025.08.005
Noy Soffer Aranov
We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the random variable which returns the degree of the smallest denominator Q, for which the ball of a fixed radius around a point contains a rational function of the form . Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and P-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.
{"title":"On the minimal denominator problem in function fields","authors":"Noy Soffer Aranov","doi":"10.1016/j.jnt.2025.08.005","DOIUrl":"10.1016/j.jnt.2025.08.005","url":null,"abstract":"<div><div>We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the random variable which returns the degree of the smallest denominator <em>Q</em>, for which the ball of a fixed radius around a point contains a rational function of the form <span><math><mfrac><mrow><mi>P</mi></mrow><mrow><mi>Q</mi></mrow></mfrac></math></span>. Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and <em>P</em>-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 35-48"},"PeriodicalIF":0.7,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049555","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 : 2025-08-22DOI: 10.1016/j.jnt.2025.07.012
Weitong Wang
We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thus, we can show some statistical results as follows. For abelian extensions over a fixed number field K, we show infinite -moments for the Sylow p-subgroup of the relative class group when p divides the degree of the extension. For sextic number fields with -closure, we can show infinite -moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field.
{"title":"Invariant part of class groups and distribution of relative class group","authors":"Weitong Wang","doi":"10.1016/j.jnt.2025.07.012","DOIUrl":"10.1016/j.jnt.2025.07.012","url":null,"abstract":"<div><div>We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thus, we can show some statistical results as follows. For abelian extensions over a fixed number field <em>K</em>, we show infinite <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-moments for the Sylow <em>p</em>-subgroup of the relative class group when <em>p</em> divides the degree of the extension. For sextic number fields with <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-closure, we can show infinite <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 691-748"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904483","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 : 2025-08-22DOI: 10.1016/j.jnt.2025.07.006
Zhan Shi , Lin Weng
For elliptic curves over rationals, there are a well-known conjecture of Sato–Tate and a new computational guided murmuration phenomenon, for which the abelian Artin zeta functions are used. In this paper, we show that both the murmurations and the Sato–Tate conjecture stand equally well for non-abelian high rank zeta functions of the p-reductions of elliptic curves over rationals. We establish our results by carefully examining asymptotic behaviors of the p-reduction invariants , the rank n analogous of the rank one a-invariant of elliptic curve . Such asymptotic results are based on a ‘counting miracle’ of the so-called - and -invariants of in rank n, and a remarkable recursive relation on the -invariants, both established by Weng–Zagier in [22].
{"title":"Murmurations and Sato–Tate conjectures for high rank zetas of elliptic curves","authors":"Zhan Shi , Lin Weng","doi":"10.1016/j.jnt.2025.07.006","DOIUrl":"10.1016/j.jnt.2025.07.006","url":null,"abstract":"<div><div>For elliptic curves over rationals, there are a well-known conjecture of Sato–Tate and a new computational guided murmuration phenomenon, for which the abelian Artin zeta functions are used. In this paper, we show that both the murmurations and the Sato–Tate conjecture stand equally well for non-abelian high rank zeta functions of the <em>p</em>-reductions of elliptic curves over rationals. We establish our results by carefully examining asymptotic behaviors of the <em>p</em>-reduction invariants <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub><mspace></mspace><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>, the rank <em>n</em> analogous of the rank one <em>a</em>-invariant <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>p</mi><mo>−</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msub></math></span> of elliptic curve <span><math><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Such asymptotic results are based on a ‘counting miracle’ of the so-called <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub></math></span>- and <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub></math></span>-invariants of <span><math><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> in rank <em>n</em>, and a remarkable recursive relation on the <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>E</mi><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>n</mi></mrow></msub></math></span>-invariants, both established by Weng–Zagier in <span><span>[22]</span></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 948-968"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144922301","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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