Journal of Number Theory最新文献

{"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":"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":"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":"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}

{"title":"On prime numbers and quadratic forms represented by positive-definite, primitive quadratic forms","authors":"Yves Martin","doi":"10.1016/j.jnt.2024.12.014","DOIUrl":"10.1016/j.jnt.2024.12.014","url":null,"abstract":"<div><div>In this note we show that every positive-definite, integral, primitive, <em>n</em>-ary quadratic form with <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> represents infinitely many prime numbers and infinitely many primitive, non-equivalent, <em>m</em>-ary quadratic forms for each <span><math><mn>2</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. We do so via an inductive argument which only requires to know the statement for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> (proved by H. Weber in 1882), and elementary linear algebra. The result on the representation of prime numbers by <em>n</em>-ary quadratic forms for arbitrary <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math></span> can be deduced from theorems already known, but the proof below is more direct and seems to be new in the literature. As an application we establish a non-vanishing result for Fourier-Jacobi coefficients of Siegel modular forms of any degree, level and Dirichlet character, subject to a condition on the conductor of the character.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 26-36"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143520773","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":"Uniform bounds for Kloosterman sums of half-integral weight, same-sign case","authors":"Qihang Sun","doi":"10.1016/j.jnt.2024.11.012","DOIUrl":"10.1016/j.jnt.2024.11.012","url":null,"abstract":"<div><div>In the previous paper <span><span>[Sun24]</span></span>, the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more precise bound for a certain class of multipliers compared to the 1983 result by Goldfeld and Sarnak and generalizes the 2009 result from Sarnak and Tsimerman to the half-integral weight case. However, the author only considered the case when the parameters satisfied <span><math><mover><mrow><mi>m</mi></mrow><mrow><mo>˜</mo></mrow></mover><mover><mrow><mi>n</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>&lt;</mo><mn>0</mn></math></span>. In this paper, we prove the same uniform bound when <span><math><mover><mrow><mi>m</mi></mrow><mrow><mo>˜</mo></mrow></mover><mover><mrow><mi>n</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>&gt;</mo><mn>0</mn></math></span> for further applications.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 104-139"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548347","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 Mahler measure of a family of polynomials with arbitrarily many variables","authors":"Siva Sankar Nair","doi":"10.1016/j.jnt.2024.11.011","DOIUrl":"10.1016/j.jnt.2024.11.011","url":null,"abstract":"<div><div>We present an exact formula for the Mahler measure of an infinite family of polynomials with arbitrarily many variables. The formula is obtained by manipulating the integral defining the Mahler measure using certain transformations, followed by an iterative process that reduces this computation to the evaluation of certain polylogarithm functions at sixth roots of unity. This yields values of the Riemann zeta function and the Dirichlet <em>L</em>-function associated to the character of conductor 3.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 214-272"},"PeriodicalIF":0.6,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637189","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":"Relative sizes of iterated sumsets","authors":"Noah Kravitz","doi":"10.1016/j.jnt.2025.01.007","DOIUrl":"10.1016/j.jnt.2025.01.007","url":null,"abstract":"<div><div>Let <em>hA</em> denote the <em>h</em>-fold sumset of a subset <em>A</em> of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, there exist finite subsets <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊆</mo><mi>Z</mi></math></span> such that for each <span><math><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mi>H</mi></math></span>, the relative order of the quantities <span><math><mo>|</mo><mi>h</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>,</mo><mo>…</mo><mo>,</mo><mo>|</mo><mi>h</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo></math></span> is given by <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>. We also establish extensions where <span><math><mi>Z</mi></math></span> is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"272 ","pages":"Pages 113-128"},"PeriodicalIF":0.6,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510120","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}