Journal of Number Theory最新文献

{"title":"Corrigendum to “On certain maximal hyperelliptic curves related to Chebyshev polynomials” [J. Number Theory 203 (2019) 276–293]","authors":"Saeed Tafazolian , Jaap Top","doi":"10.1016/j.jnt.2024.10.011","DOIUrl":"10.1016/j.jnt.2024.10.011","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 427-428"},"PeriodicalIF":0.6,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744084","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":"Period of the Ikeda-Miyawaki lift","authors":"Tomoyoshi Ibukiyama ,&nbsp;Hidenori Katsurada ,&nbsp;Hisashi Kojima","doi":"10.1016/j.jnt.2024.09.014","DOIUrl":"10.1016/j.jnt.2024.09.014","url":null,"abstract":"<div><div>In this paper, first we give a weak version of Ikeda's conjecture on the period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously in some cases.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 341-369"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744086","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":"Rational configuration problems and a family of curves","authors":"Jonathan Love","doi":"10.1016/j.jnt.2024.09.008","DOIUrl":"10.1016/j.jnt.2024.09.008","url":null,"abstract":"<div><div>Given <figure><img></figure>, we consider the number of rational points on the genus one curve<span><span><span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><mi>b</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>c</mi><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></math></span></span></span> We prove that the set of <em>η</em> for which <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo><mo>≠</mo><mo>∅</mo></math></span> has density zero, and that if a rational point <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>∈</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> exists, then <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> is infinite unless a certain explicit polynomial in <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> vanishes.</div><div>Curves of the form <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>η</mi></mrow></msub></math></span> naturally occur in the study of configurations of points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> passes through a rational point on the unit circle, then it contains a dense set of points <em>P</em> such that the distances from <em>P</em> to each of the three points <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> are all rational. We also prove some results regarding whether a rational number can be expressed as a sum or product of slopes of rational right triangles.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 370-396"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On gamma factors of Rankin–Selberg integrals for U2ℓ × ResE/FGLn","authors":"Kazuki Morimoto","doi":"10.1016/j.jnt.2024.09.013","DOIUrl":"10.1016/j.jnt.2024.09.013","url":null,"abstract":"<div><div>In this paper, we prove the fundamental properties of gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of generic representations <span><math><mo>(</mo><mi>π</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>E</mi><mo>)</mo></math></span> for a local field <em>F</em> of characteristic zero and a quadratic extension <em>E</em> of <em>F</em>. We also prove similar results for pairs of generic representations <span><math><mo>(</mo><mi>π</mi><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊗</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>×</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span>. As a corollary, we prove that the gamma factors arising from Langlands–Shahidi method and our gamma factors coincide.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 203-246"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744090","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 height of some generators of Galois extensions with big Galois group","authors":"Jonathan Jenvrin","doi":"10.1016/j.jnt.2024.10.004","DOIUrl":"10.1016/j.jnt.2024.10.004","url":null,"abstract":"<div><div>We study the height of generators of Galois extensions of the rationals having the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height tends to infinity as <em>n</em> increases. This provides an analogue of a result by Amoroso, originally established for the symmetric group.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 78-105"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744205","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 generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values","authors":"Annika Burmester","doi":"10.1016/j.jnt.2024.09.011","DOIUrl":"10.1016/j.jnt.2024.09.011","url":null,"abstract":"<div><div>We present the <em>τ</em>-invariant balanced quasi-shuffle algebra <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span>, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple <em>q</em>-zeta values. In particular, <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span> has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span> of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span>. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span> onto <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span>. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit <span><math><mi>q</mi><mo>→</mo><mn>1</mn></math></span> of multiple <em>q</em>-zeta values.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 106-137"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uniform boundedness on rational maps with automorphisms","authors":"Minsik Han","doi":"10.1016/j.jnt.2024.09.012","DOIUrl":"10.1016/j.jnt.2024.09.012","url":null,"abstract":"<div><div>In this paper, we study the dynamical uniform boundedness conjecture over a family of rational maps with certain nontrivial automorphisms. Specifically, we consider a family of rational maps of an arbitrary degree <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span> whose automorphism group contains the cyclic group of order <em>d</em>. We prove that a subfamily of this family satisfies the dynamical uniform boundedness conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 138-156"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744127","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 actions on Tate modules of Abelian varieties with semi-stable reduction","authors":"Khai-Hoan Nguyen-Dang","doi":"10.1016/j.jnt.2024.08.008","DOIUrl":"10.1016/j.jnt.2024.08.008","url":null,"abstract":"<div><div>Let <em>p</em> be a rational prime number, let <em>K</em> denote a finite extension of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, <span><math><mover><mrow><mi>K</mi></mrow><mo>‾</mo></mover></math></span> some fixed algebraic closure of <em>K</em>. Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> be the absolute Galois group of <em>K</em> and let <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>⊂</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> be its inertial subgroup. Let <em>A</em> be an Abelian variety defined over <em>K</em>, with semi-stable reduction. In this note, we give a criterion for which <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub></mrow></msup><mo>=</mo><mn>0</mn></math></span>, where <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is the <em>p</em>-adic Tate module associated to <em>A</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 247-259"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744085","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":"First order Stickelberger modules over imaginary quadratic fields","authors":"Saad El Boukhari","doi":"10.1016/j.jnt.2024.10.005","DOIUrl":"10.1016/j.jnt.2024.10.005","url":null,"abstract":"<div><div>Let <span><math><mi>K</mi><mo>/</mo><mi>k</mi></math></span> be a finite abelian extension of number fields of Galois group <em>G</em> with <em>k</em> imaginary quadratic. Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a rational integer, and for a certain finite set <em>S</em> of places of <em>k</em>, let <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>S</mi></mrow></msub></math></span> be the ring of <em>S</em>-integers of <em>K</em>. We use generalized Stark elements to construct first order Stickelberger modules in odd higher algebraic <em>K</em>-groups of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>S</mi></mrow></msub></math></span>. We show that the Fitting ideal (resp. index) of these modules inside the corresponding odd <em>K</em>-groups is exactly the Fitting ideal (resp. cardinality) of the even higher algebraic <em>K</em>-group <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi><mo>,</mo><mi>S</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 1-16"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744126","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":"Counting wild quartics with prescribed discriminant and Galois closure group","authors":"Sebastian Monnet","doi":"10.1016/j.jnt.2024.10.008","DOIUrl":"10.1016/j.jnt.2024.10.008","url":null,"abstract":"<div><div>Given a 2-adic field <em>K</em>, we give a formula for the number of totally ramified quartic field extensions <span><math><mi>L</mi><mo>/</mo><mi>K</mi></math></span> with a given discriminant valuation and Galois closure group. We use these formulae to prove refinements of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 157-202"},"PeriodicalIF":0.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}