Journal of Number Theory最新文献

{"title":"Transitive actions and purely periodic N-expansions","authors":"Mohammad Javaheri","doi":"10.1016/j.jnt.2025.11.013","DOIUrl":"10.1016/j.jnt.2025.11.013","url":null,"abstract":"<div><div>We study the dynamical properties of the semigroup <span><math><mo>〈</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>〉</mo></math></span> generated by the maps <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>N</mi><mo>/</mo><mi>x</mi></math></span> acting on the positive rational numbers <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, with a focus on their connection to <em>N</em>-expansions. For <span><math><mi>N</mi><mo>∈</mo><mi>N</mi><mo>∖</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>}</mo></math></span>, we establish the existence of an infinite subset <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> on which <span><math><mo>〈</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>〉</mo></math></span> acts transitively: for any <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>A</mi></math></span>, there exists <span><math><mi>σ</mi><mo>∈</mo><mo>〈</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>〉</mo></math></span> such that <span><math><mi>y</mi><mo>=</mo><mi>σ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. We use this result to show that, for the same values of <em>N</em>, there are infinitely many rational numbers with purely periodic <em>N</em>-expansions of any sufficiently large minimal period.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 1-14"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145876998","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 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>&lt;</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":"2026-06-01","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}

{"title":"A note on mod-p local-global compatibility via Scholze's functor","authors":"Kegang Liu ,&nbsp;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":"2026-06-01","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}

{"title":"The prime number theorem over integers of power-free polynomial values","authors":"Biao Wang ,&nbsp;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":"2026-06-01","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}

{"title":"Local monodromy of 1-dimensional p-divisible groups","authors":"Tristan Phillips","doi":"10.1016/j.jnt.2025.11.008","DOIUrl":"10.1016/j.jnt.2025.11.008","url":null,"abstract":"<div><div>Let <em>G</em> be a <em>p</em>-divisible group over a complete discrete valuation ring <em>R</em> of characteristic <em>p</em>. The generic fiber of <em>G</em> determines a Galois representation <em>ρ</em>. The image of <em>ρ</em> admits a ramification filtration and a Lie filtration. We relate these filtrations in the case <em>G</em> is one dimensional, giving an equicharacteristic version of Sen's theorem in this setting. This result generalizes a result of Gross. Additionally, we prove that the representation associated to the étale part of <em>G</em> is irreducible, generalizing a result of Chai.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 259-284"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023476","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":"Contour integrations and parity results of cyclotomic Euler sums and multiple polylogarithm function","authors":"Hongyuan Rui ,&nbsp;Ce Xu","doi":"10.1016/j.jnt.2025.11.012","DOIUrl":"10.1016/j.jnt.2025.11.012","url":null,"abstract":"<div><div>In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We will prove parity results for cyclotomic Euler sums of arbitrary degree, explicit formulas for the parity of cyclotomic linear and quadratic Euler sums, as well as some formulas for the parity of cyclotomic cubic Euler sums and multiple polylogarithms. As a direct corollary, we derive known formulas concerning the parity of classical Euler sums and alternating Euler sums.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 64-96"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927958","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":"Low lying zeros of Rankin-Selberg L-functions","authors":"Alexander Shashkov","doi":"10.1016/j.jnt.2025.11.009","DOIUrl":"10.1016/j.jnt.2025.11.009","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We study the low lying zeros of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; Rankin-Selberg &lt;em&gt;L&lt;/em&gt;-functions. Assuming the Generalized Riemann Hypothesis, we compute the 1-level density of the low-lying zeroes of &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; averaged over families of Rankin-Selberg convolutions, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are cuspidal newforms with even weights &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and prime levels &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, respectively. The Katz-Sarnak density conjecture predicts that in the limit, the 1-level density of suitable families of &lt;em&gt;L&lt;/em&gt;-functions is the same as the distribution of eigenvalues of corresponding families of random matrices. The 1-level density relies on a smooth test function &lt;em&gt;ϕ&lt;/em&gt; whose Fourier transform &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; has compact support. In general, we show the Katz-Sarnak density conjecture holds for test functions &lt;em&gt;ϕ&lt;/em&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;supp&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. When &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, we prove the density conjecture for &lt;span&gt;&lt;math&gt;&lt;mi&gt;supp&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; when &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;supp&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; when &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. A secondary term contributes to the 1-level density when the support of &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; ex","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"283 ","pages":"Pages 170-215"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927961","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":"Corrigendum to “Divisibility of the multiplicative order modulo monic irreducible polynomials over finite fields” [J. Number Theory 277 (2025) 105–123]","authors":"Joaquim Cera Da Conceição","doi":"10.1016/j.jnt.2025.11.002","DOIUrl":"10.1016/j.jnt.2025.11.002","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 144-146"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737657","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 ideal class groups of totally degenerate number rings","authors":"Ruben Hambardzumyan ,&nbsp;Mihran Papikian","doi":"10.1016/j.jnt.2025.11.001","DOIUrl":"10.1016/j.jnt.2025.11.001","url":null,"abstract":"<div><div>Let <span><math><mi>χ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring <span><math><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo><mo>/</mo><mo>(</mo><mi>χ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span>. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of <span><math><mi>χ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of <span><math><mi>χ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is 2 or 3.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 118-143"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737656","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 non-vanishing of Hilbert Poincaré series","authors":"Mingkuan Zhang ,&nbsp;Yichao Zhang","doi":"10.1016/j.jnt.2025.11.003","DOIUrl":"10.1016/j.jnt.2025.11.003","url":null,"abstract":"<div><div>We prove that if <em>ν</em> has small norm with respect to the level and the weight, the <em>ν</em>-th Hilbert Poincaré series does not vanish identically. We also prove Selberg's identity on Kloosterman sums in the case of number fields, which implies certain vanishing and non-vanishing relations of Hilbert Poincaré series when the narrow class number is 1. Finally, we pass to the adelic setting and interpret the problem via Hecke operators.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"282 ","pages":"Pages 31-47"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733459","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}