S. Herrero , Ö. Imamoḡlu , A.-M. von Pippich , M. Schwagenscheidt
{"title":"Twisted traces of modular functions on hyperbolic 3-space","authors":"S. Herrero , Ö. Imamoḡlu , A.-M. von Pippich , M. Schwagenscheidt","doi":"10.1016/j.jnt.2025.06.010","DOIUrl":"10.1016/j.jnt.2025.06.010","url":null,"abstract":"<div><div>We compute analogues of twisted traces of CM values of harmonic modular functions on hyperbolic 3-space and show that they are essentially given by Fourier coefficients of the <em>j</em>-invariant. From this we deduce that the twisted traces of these harmonic modular functions are integers. Additionally, we compute the twisted traces of Eisenstein series on hyperbolic 3-space in terms of Dirichlet <em>L</em>-functions and divisor sums.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 154-169"},"PeriodicalIF":0.6,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687067","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":"Archimedean Bump-Friedberg integrals on GL(4,R)","authors":"Miki Hirano , Taku Ishii , Tadashi Miyazaki","doi":"10.1016/j.jnt.2025.06.017","DOIUrl":"10.1016/j.jnt.2025.06.017","url":null,"abstract":"<div><div>As an application of explicit formulas of Whittaker functions on <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mn>4</mn><mo>,</mo><mi>R</mi><mo>)</mo></math></span>, we determine test vectors for Bump-Friedberg integrals for all generic representations of <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mn>4</mn><mo>,</mo><mi>R</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 411-456"},"PeriodicalIF":0.7,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144723682","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":"Replenishing p-adic L-functions and Euler systems at their bad primes","authors":"Raiza Corpuz , Daniel Delbourgo","doi":"10.1016/j.jnt.2025.06.011","DOIUrl":"10.1016/j.jnt.2025.06.011","url":null,"abstract":"<div><div>Let <em>M</em> denote a pure motive over a totally real number field <em>F</em>. If the Euler factors associated to the motive <em>M</em> at its bad primes satisfy a certain condition (<span><math><msup><mrow><mi>S</mi></mrow><mrow><mtext>dep</mtext></mrow></msup></math></span>), then we show that the (conjectural) primitive <em>p</em>-adic <em>L</em>-function attached to <em>M</em> is a linear combination of two <em>S</em>-depleted versions. We next verify (<span><math><msup><mrow><mi>S</mi></mrow><mrow><mtext>dep</mtext></mrow></msup></math></span>) holds for single, double and triple products of Hilbert modular forms, as well as their symmetric squares. If <span><math><mi>F</mi><mo>=</mo><mi>Q</mi></math></span> and each <em>ℓ</em>-adic realisation <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mtext>ét</mtext></mrow></msubsup><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is self-dual, this factorisation yields <em>primitive</em> zeta-elements attached to <span><math><msub><mrow><mi>M</mi></mrow><mrow><mo>/</mo><msup><mrow><mi>Q</mi></mrow><mrow><mtext>cyc</mtext></mrow></msup></mrow></msub></math></span> and a sharper divisibility in the Iwasawa Main Conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 527-563"},"PeriodicalIF":0.7,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724476","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}
Stefan Barańczuk , Bartosz Naskręcki , Matteo Verzobio
{"title":"Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve","authors":"Stefan Barańczuk , Bartosz Naskręcki , Matteo Verzobio","doi":"10.1016/j.jnt.2025.06.001","DOIUrl":"10.1016/j.jnt.2025.06.001","url":null,"abstract":"<div><div>Let <em>A</em> be an abelian variety defined over a number field <em>K</em>, <span><math><mi>E</mi><mo>/</mo><mi>K</mi></math></span> be an elliptic curve, and <span><math><mi>ϕ</mi><mo>:</mo><mi>A</mi><mo>→</mo><msup><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> be an isogeny defined over <em>K</em>. Let <span><math><mi>P</mi><mo>∈</mo><mi>A</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span> be such that <span><math><mi>ϕ</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>Rank</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo>(</mo><mo>〈</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>〉</mo><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We will study a divisibility sequence related to the point <em>P</em> and show its relation with elliptic divisibility sequences.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 170-183"},"PeriodicalIF":0.6,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687068","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 converse theorem for quasi-split even special orthogonal groups over finite fields","authors":"Alexander Hazeltine","doi":"10.1016/j.jnt.2025.06.006","DOIUrl":"10.1016/j.jnt.2025.06.006","url":null,"abstract":"<div><div>We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties that arise from the outer automorphism and non-split part of the torus. The outer automorphism is handled similarly to the split case, while new ideas are developed to overcome the non-split part of the torus.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 479-511"},"PeriodicalIF":0.7,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144720914","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":"Bogomolov property and Galois representations","authors":"Francesco Amoroso , Lea Terracini","doi":"10.1016/j.jnt.2025.06.002","DOIUrl":"10.1016/j.jnt.2025.06.002","url":null,"abstract":"<div><div>In 2013 P. Habegger proved the Bogomolov property for the field generated over <span><math><mi>Q</mi></math></span> by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions cut out by Galois representations arising from more general modular forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 294-322"},"PeriodicalIF":0.6,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144712952","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 p-adic modularity in the p-adic Heisenberg algebra","authors":"Cameron Franc , Geoffrey Mason","doi":"10.1016/j.jnt.2025.05.017","DOIUrl":"10.1016/j.jnt.2025.05.017","url":null,"abstract":"<div><div>We establish existence theorems for the image of the normalized character map of the <em>p</em>-adic Heisenberg algebra <em>S</em> taking values in the algebra of Serre <em>p</em>-adic modular forms <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. In particular, we describe the construction of an analytic family of states in <em>S</em> whose character values are the well-known Λ-adic family of <em>p</em>-adic Eisenstein series of level one built from classical Eisenstein series. This extends previous work treating a specialization at weight 2, and illustrates that the image of the character map contains nonzero <em>p</em>-adic modular forms of every <em>p</em>-adic weight. In a different direction, we prove that for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> the image of the rescaled character map contains every overconvergent 2-adic modular form of weight zero and tame level one; in particular, it contains the polynomial algebra <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><msup><mrow><mi>j</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span>. For general primes <em>p</em>, we study the square-bracket formalism for <em>S</em> and develop the idea that although states in <em>S</em> do not generally have a conformal weight, they can acquire a <em>p</em>-adic weight in the sense of Serre.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"279 ","pages":"Pages 117-153"},"PeriodicalIF":0.6,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144614484","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}
Eleni Agathocleous , Antoine Joux , Daniele Taufer
{"title":"Elliptic curves over Hasse pairs","authors":"Eleni Agathocleous , Antoine Joux , Daniele Taufer","doi":"10.1016/j.jnt.2025.05.008","DOIUrl":"10.1016/j.jnt.2025.05.008","url":null,"abstract":"<div><div>We call a pair of distinct prime powers <span><math><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>)</mo></math></span> a Hasse pair if <span><math><mo>|</mo><msqrt><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msqrt><mo>−</mo><msqrt><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msqrt><mo>|</mo><mo>≤</mo><mn>1</mn></math></span>. For such pairs, we study the relation between the set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of isomorphism classes of elliptic curves defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></math></span> with <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> points, and the set <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of isomorphism classes of elliptic curves over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> with <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> points. When both families <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> contain only ordinary elliptic curves, we prove that their isogeny graphs are isomorphic. When supersingular curves are involved, we describe which curves might belong to these sets. We also show that if both the <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are odd and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≠</mo><mo>∅</mo></math></span>, then <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> always contains an ordinary elliptic curve. Conversely, if <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is even, then <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> may contain only supersingular curves precisely when <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is a given power of a Fermat or a Mersenne prime. In the case of odd Hasse pairs, we could not rule out the possibility of an empty union <span><math><msub><mrow><mi>E</mi></mrow","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 924-952"},"PeriodicalIF":0.6,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491596","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":"Equidistribution of Hecke orbits on the Picard group of definite Shimura curves","authors":"Matias Alvarado, Patricio Pérez-Piña","doi":"10.1016/j.jnt.2025.05.002","DOIUrl":"10.1016/j.jnt.2025.05.002","url":null,"abstract":"<div><div>We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular Drinfeld modules of rank 2. Our approach is via the automorphic method, using bounds for coefficients of cuspidal automorphic forms of Drinfeld type as the main tool.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 715-725"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321677","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}
Santiago Arango-Piñeros , María Chara , Asimina S. Hamakiotes , Kiran S. Kedlaya , Gustavo Rama
{"title":"Bounds for the relative class number problem for function fields","authors":"Santiago Arango-Piñeros , María Chara , Asimina S. Hamakiotes , Kiran S. Kedlaya , Gustavo Rama","doi":"10.1016/j.jnt.2025.05.010","DOIUrl":"10.1016/j.jnt.2025.05.010","url":null,"abstract":"<div><div>We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 977-1010"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522824","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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