Journal of Number Theory最新文献

{"title":"Asymptotic solutions of the generalized Fermat-type equation of signature (p,p,3) over totally real number fields","authors":"Satyabrat Sahoo ,&nbsp;Narasimha Kumar","doi":"10.1016/j.jnt.2025.01.020","DOIUrl":"10.1016/j.jnt.2025.01.020","url":null,"abstract":"<div><div>In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>p</mi><mo>,</mo><mn>3</mn><mo>)</mo></math></span> over totally real number fields <em>K</em>, i.e., <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with prime exponent <em>p</em> and <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. For certain class of fields <em>K</em>, we prove that <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> has no asymptotic solutions over <em>K</em> (resp., solutions of certain type over <em>K</em>) with restrictions on <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span> (resp., for all <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>). Finally, we present several local criteria over <em>K</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 56-71"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529155","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":"Locally induced Galois representations with exceptional residual images","authors":"Chengyang Bao","doi":"10.1016/j.jnt.2024.12.015","DOIUrl":"10.1016/j.jnt.2024.12.015","url":null,"abstract":"<div><div>In this paper, we classify all continuous Galois representations <span><math><mi>ρ</mi><mo>:</mo><mrow><mi>Gal</mi></mrow><mo>(</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover><mo>/</mo><mi>Q</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> which are unramified outside <span><math><mo>{</mo><mi>p</mi><mo>,</mo><mo>∞</mo><mo>}</mo></math></span> and locally induced at <em>p</em>, under the assumption that <span><math><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></math></span> is exceptional, that is, has image of order prime to <em>p</em>. We prove two results. If <em>f</em> is a level one cuspidal eigenform and one of the <em>p</em>-adic Galois representations <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> associated to <em>f</em> has exceptional residual image, then <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> is not locally induced and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>≠</mo><mn>0</mn></math></span>. If <em>ρ</em> is locally induced at <em>p</em> and with exceptional residual image, and furthermore certain subfields of the fixed field of the kernel of <span><math><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></math></span> are assumed to have class numbers prime to <em>p</em>, then <em>ρ</em> has finite image up to a twist.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 49-66"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552300","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 the factor rings of Z[n3]","authors":"Tomasz Jędrzejak","doi":"10.1016/j.jnt.2025.01.022","DOIUrl":"10.1016/j.jnt.2025.01.022","url":null,"abstract":"<div><div>We give a description of the structure of factor rings of <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow></math></span> where (without loss of generality) <em>n</em> is a positive integer which is not a cube. For example, we prove that <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>+</mo><mi>c</mi><mroot><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mroot><mo>)</mo></mrow></math></span> is isomorphic to the ring of integers modulo <span><math><mo>|</mo><mi>N</mi><mo>|</mo></math></span>, if <span><math><mi>gcd</mi><mo>⁡</mo><mrow><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>a</mi><mi>c</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></span> where <span><math><mi>N</mi><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>n</mi><msup><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><mi>n</mi><mi>a</mi><mi>b</mi><mi>c</mi></math></span> is the norm of the generator. We also characterize the structure of these factor rings for others integers <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></math></span>. Finally, we describe <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow><mo>/</mo><mi>I</mi></math></span> for certain non-principal ideals <em>I</em>. We also present many corollaries regarding irreducible and prime elements in <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow></math></span> and give numerous examples.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 104-118"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563378","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 torsion subgroups of elliptic curves over quartic, quintic and sextic number fields","authors":"Mustafa Umut Kazancıoğlu,&nbsp;Mohammad Sadek","doi":"10.1016/j.jnt.2025.01.017","DOIUrl":"10.1016/j.jnt.2025.01.017","url":null,"abstract":"<div><div>The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree <em>d</em>, <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span>, is not completely determined. However, the list of groups <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>d</mi><mo>)</mo></math></span>, <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span>, that can be realized as torsion subgroups for infinitely many non-isomorphic elliptic curves over these fields is known. We address the question of which torsion subgroups can arise over a given number field of degree <em>d</em>. In fact, given <span><math><mi>G</mi><mo>∈</mo><msup><mrow><mi>Φ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>d</mi><mo>)</mo></math></span> and a number field <em>K</em> of degree <em>d</em>, we give explicit criteria telling whether <em>G</em> is realized finitely or infinitely often over <em>K</em>. We also give results on the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with torsion <em>G</em>. Finally, we give examples of number fields <em>K</em> of degree <em>d</em>, <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span>, over which the Mordell-Weil rank of elliptic curves with prescribed torsion is bounded from above.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 37-55"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529156","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 counting function for Elkies primes","authors":"Meher Elijah Lippmann ,&nbsp;Kevin J. McGown","doi":"10.1016/j.jnt.2024.12.009","DOIUrl":"10.1016/j.jnt.2024.12.009","url":null,"abstract":"<div><div>Let <em>E</em> be an elliptic curve over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> where <em>q</em> is a prime power. The Schoof–Elkies–Atkin (SEA) algorithm is a standard method for counting the number of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-points on <em>E</em>. The asymptotic complexity of the SEA algorithm depends on the distribution of the so-called Elkies primes.</div><div>Assuming GRH, we prove that the least Elkies prime is bounded by <span><math><msup><mrow><mo>(</mo><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>4</mn><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> when <span><math><mi>q</mi><mo>≥</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>9</mn></mrow></msup></math></span>. Previously, Satoh and Galbraith established an upper bound of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>. Let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> denote the number of Elkies primes less than <em>X</em>. Assuming GRH, we also show<span><span><span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>π</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mfrac><mrow><msqrt><mrow><mi>X</mi></mrow></msqrt><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>q</mi><mi>X</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>X</mi></mrow></mfrac><mo>)</mo></mrow><mspace></mspace><mo>.</mo></math></span></span></span></div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 35-48"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552301","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":"A polytopal generalization of Apollonian packings and Descartes' theorem","authors":"Jorge L. Ramírez Alfonsín ,&nbsp;Iván Rasskin","doi":"10.1016/j.jnt.2024.11.010","DOIUrl":"10.1016/j.jnt.2024.11.010","url":null,"abstract":"<div><div>We present a generalization of Descartes' theorem for the family of polytopal sphere packings arising from uniform polytopes. The corresponding quadratic equation is expressed in terms of geometric invariants of uniform polytopes which are closely connected to canonical realizations of edge-scribable polytopes. We use our generalization to construct integral Apollonian packings based on the Platonic solids. Additionally, we also introduce and discuss a new spectral invariant for edge-scribable polytopes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 67-103"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552302","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":"Semisimple Langlands for GL2(Qp) and mod p Hecke modules","authors":"Cédric Pépin ,&nbsp;Tobias Schmidt","doi":"10.1016/j.jnt.2024.11.013","DOIUrl":"10.1016/j.jnt.2024.11.013","url":null,"abstract":"<div><div>Let <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span> and let <span><math><mi>Z</mi><mo>(</mo><msubsup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>)</mo></math></span> be the centre of the mod <em>p</em> pro-<em>p</em>-Iwahori Hecke algebra of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>. Let <em>X</em> be the projective curve parametrizing 2-dimensional mod <em>p</em> semi-simple representations of the absolute Galois group <span><math><mrow><mi>Gal</mi></mrow><mo>(</mo><msub><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>/</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>. We construct a quotient morphism of schemes <span><math><mi>L</mi><mo>:</mo><mi>Spec</mi><mspace></mspace><mi>Z</mi><mo>(</mo><msubsup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>)</mo><mo>→</mo><mi>X</mi></math></span>. We then show that the correspondence between the specialization <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mi>z</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></math></span> of the spherical <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></math></span>-module <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></math></span> from <span><span>[PS]</span></span> in closed points <span><math><mi>z</mi><mo>∈</mo><mi>Spec</mi><mspace></mspace><mi>Z</mi><mo>(</mo><msubsup><mrow><mi>H</mi></mrow><mrow><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>)</mo></math></span> and the Galois representation <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>L</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></msub></math></span> <em>is</em> the semi-simple mod <em>p</em> local Langlands correspondence for the group <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 219-251"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548218","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":"Complex numbers with a prescribed order of approximation and Zaremba's conjecture","authors":"Gerardo González Robert,&nbsp;Mumtaz Hussain,&nbsp;Nikita Shulga","doi":"10.1016/j.jnt.2024.12.010","DOIUrl":"10.1016/j.jnt.2024.12.010","url":null,"abstract":"<div><div>Given <span><math><mi>b</mi><mo>=</mo><mo>−</mo><mi>A</mi><mo>±</mo><mi>i</mi></math></span> with <em>A</em> being a positive integer, we can represent any complex number as a power series in <em>b</em> with coefficients in <span><math><mi>A</mi><mo>=</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>}</mo></math></span>. We prove that, for any real <span><math><mi>τ</mi><mo>≥</mo><mn>2</mn></math></span> and any non-empty proper subset <span><math><mi>J</mi><mo>(</mo><mi>b</mi><mo>)</mo></math></span> of <span><math><mi>A</mi></math></span> with at least two elements, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as power series in <em>b</em> with coefficients in <span><math><mi>J</mi><mo>(</mo><mi>b</mi><mo>)</mo></math></span> and with the irrationality exponent (in terms of Gaussian integers) equal to <em>τ</em>. One of the key ingredients in our construction is the ‘Folding Lemma’ applied to Hurwitz continued fractions. This motivates a Hurwitz continued fraction analogue of the well-known Zaremba's conjecture. We prove several results in support of this conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 1-25"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143520772","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":"Portraits of quadratic rational maps with a small critical cycle","authors":"Tyler Dunaisky ,&nbsp;David Krumm","doi":"10.1016/j.jnt.2024.12.008","DOIUrl":"10.1016/j.jnt.2024.12.008","url":null,"abstract":"<div><div>Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic critical point of period <em>n</em>, where <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></math></span>. In particular, we provide a conjecturally complete list of possible graphs of rational pre-periodic points in the case <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span>, analogous to well-known work of Poonen for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>, and we strengthen earlier results of Canci and Vishkautsan for <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span>. In addition, we address the problem of determining the representability of a given graph in our list by infinitely many distinct linear conjugacy classes of maps.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 135-159"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563377","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":"Some symbolic dynamics in real quadratic fields with applications to inhomogeneous minima","authors":"Nick Ramsey","doi":"10.1016/j.jnt.2025.01.019","DOIUrl":"10.1016/j.jnt.2025.01.019","url":null,"abstract":"<div><div>Let <em>K</em> be a real quadratic field. We use a symbolic coding of the action of a fundamental unit on the torus <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>=</mo><mo>(</mo><mi>K</mi><msub><mrow><mo>⊗</mo></mrow><mrow><mi>Q</mi></mrow></msub><mi>R</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> to study the family of subsets <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⊆</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> of norm distance ≥<em>t</em> from the origin. As an application, we prove that inhomogeneous spectrum of <em>K</em> contains a dense set of elements of <em>K</em>, and conclude that all isolated inhomogeneous minima lie in <em>K</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 119-134"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563379","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}