{"title":"Delone sets associated with badly approximable triangles","authors":"Shigeki Akiyama , Emily R. Korfanty , Yan-li Xu","doi":"10.1016/j.jnt.2025.04.004","DOIUrl":"10.1016/j.jnt.2025.04.004","url":null,"abstract":"<div><div>We construct new Delone sets associated with badly approximable numbers which are expected to have rotationally invariant diffraction. We optimize the discrepancy of corresponding tile orientations by investigating the linear equation <span><math><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>1</mn></math></span> where <em>πx</em>, <em>πy</em>, <em>πz</em> are three angles of a triangle used in the construction and <em>x</em>, <em>y</em>, <em>z</em> are badly approximable. In particular, we show that there are exactly two solutions that have the smallest partial quotients by lexicographical ordering.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 285-316"},"PeriodicalIF":0.6,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263318","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}
Leon Chou , Summer Haag , Jake Huryn , Andrew Ledoan
{"title":"The error term in counting prime pairs","authors":"Leon Chou , Summer Haag , Jake Huryn , Andrew Ledoan","doi":"10.1016/j.jnt.2025.04.009","DOIUrl":"10.1016/j.jnt.2025.04.009","url":null,"abstract":"<div><div>We relate the size of the error term in the Hardy–Littlewood conjectured formula for the number of prime pairs to the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm of an exponential sum over the primes formed with the von Mangoldt function.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 422-450"},"PeriodicalIF":0.6,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270782","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":"Modularity of elliptic curves over cyclotomic Zp-extensions of real quadratic fields","authors":"Sho Yoshikawa","doi":"10.1016/j.jnt.2025.04.018","DOIUrl":"10.1016/j.jnt.2025.04.018","url":null,"abstract":"<div><div>We prove that all elliptic curves defined over the cyclotomic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted <em>L</em>-function is a <em>p</em>-adic unit. Our result is a refinement of a result of X. Zhang, which is a real quadratic analogue of a result of Thorne.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 510-526"},"PeriodicalIF":0.6,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272596","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":"Fields generated by points on superelliptic curves","authors":"Lea Beneish , Christopher Keyes","doi":"10.1016/j.jnt.2025.04.011","DOIUrl":"10.1016/j.jnt.2025.04.011","url":null,"abstract":"<div><div>We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over <span><math><mi>Q</mi></math></span> with fixed degree <em>n</em> and discriminant bounded by <em>X</em>. For <em>C</em> a fixed such curve given by an affine equation <span><math><msup><mrow><mi>y</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> where <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>d</mi><mo>=</mo><mi>deg</mi><mo></mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≥</mo><mi>m</mi></math></span>, we find that for all degrees <em>n</em> divisible by <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>m</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> and sufficiently large, the number of such fields is asymptotically bounded below by <span><math><msup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup></math></span>, where <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. We then give geometric heuristics suggesting that for n not divisible by <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>m</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>, degree <em>n</em> points may be less abundant than those for which <em>n</em> is divisible by <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>m</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span> and provide an example of conditions under which a curve is known to have finitely many points of certain degrees.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 380-421"},"PeriodicalIF":0.6,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263277","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 asymptotics for the shifted convolutions of a class of multiplicative functions","authors":"Chengchao Huang , Guangshi Lü","doi":"10.1016/j.jnt.2025.04.017","DOIUrl":"10.1016/j.jnt.2025.04.017","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be a multiplicative function and its Dirichlet series <span><math><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mfrac><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></mfrac></math></span> satisfies three reasonable conditions. We establish an asymptotic formula of shifted sum of the form<span><span><span><math><munder><mo>∑</mo><mrow><mi>X</mi><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>2</mn><mi>X</mi></mrow></munder><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mover><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>h</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span></span></span>for almost all <span><math><mi>h</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>H</mi><mo>]</mo></math></span>, provided that <span><math><msup><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>ε</mi></mrow></msup><mo>≤</mo><mi>H</mi><mo>≤</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>ε</mi></mrow></msup></math></span>, where <span><math><mn>0</mn><mo>≤</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mn>1</mn></math></span> is a constant that is only related to <span><math><mi>F</mi></math></span>. Let <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> be normalized Hecke eigenvalues of a holomorphic cusp form <em>f</em>. As applications, we consider a few special cases, including <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>⁎</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>⁎</mo><mo>⋯</mo><mo>⁎</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> is the <em>i</em>-fold convolution of <s","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 128-177"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254316","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":"Upper bounds for moments of zeta sums","authors":"Peng Gao","doi":"10.1016/j.jnt.2025.04.002","DOIUrl":"10.1016/j.jnt.2025.04.002","url":null,"abstract":"<div><div>We establish upper bounds for moments of zeta sums using results on shifted moments of the Riemann zeta function under the Riemann hypothesis.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 47-63"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144243164","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":"Newton polygons for certain two variable exponential sums","authors":"Bolun Wei","doi":"10.1016/j.jnt.2025.04.005","DOIUrl":"10.1016/j.jnt.2025.04.005","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>y</mi><mo>+</mo><mfrac><mrow><mi>t</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></math></span> be a Laurent polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with <em>t</em> a parameter. This paper studies the Newton polygon for the <em>L</em>-function <span><math><mi>L</mi><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><mi>T</mi><mo>)</mo></math></span> of toric exponential sums attached to <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> over a finite field with characteristic <em>p</em>. The explicit Newton polygon is obtained by systematically using Dwork's <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-splitting function with an appropriate choice of basis for cohomology following the method of <span><span>[2]</span></span>. Our result provides a nontrivial explicit Newton polygon for a non-ordinary family of more than one variable with asymptotical behavior, which gives evidence of Wan's limit conjecture <span><span>[16]</span></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 178-213"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144243507","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":"Three-torsion subgroups and wild conductors of genus 3 hyperelliptic curves","authors":"Elvira Lupoian","doi":"10.1016/j.jnt.2025.04.015","DOIUrl":"10.1016/j.jnt.2025.04.015","url":null,"abstract":"<div><div>We give a practical method for computing the 3-torsion subgroup of the Jacobian of a genus 3 hyperelliptic curve. We define a scheme for the 3-torsion points of the Jacobian and use complex approximations, homotopy continuation and lattice reduction to find precise expression for the 3-torsion. In the latter stages of the paper we explain how the 3-torsion subgroup can be used to compute the wild part of the local exponent of the conductor at 2.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 267-284"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254318","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":"Imaginary quadratic fields F with X0(15)(F) finite","authors":"Tim Evink","doi":"10.1016/j.jnt.2025.04.008","DOIUrl":"10.1016/j.jnt.2025.04.008","url":null,"abstract":"<div><div>Caraiani and Newton have proven that if <em>F</em> is an imaginary quadratic number field such that <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>15</mn><mo>)</mo></math></span> has rank 0 over <em>F</em>, then every elliptic curve over <em>F</em> is modular. This paper is concerned with the quadratic fields <span><math><mi>F</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mi>p</mi></mrow></msqrt><mo>)</mo></math></span> for a prime number <em>p</em>. We give explicit conditions on <em>p</em> under which the rank is 0, and prove that these conditions are satisfied for 87.5% of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank 0 follows from a 4-descent over <span><math><mi>Q</mi></math></span> on the quadratic twist <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow><mrow><mo>−</mo><mi>p</mi></mrow></msub></math></span>. To prove this, we perform two consecutive 2-descents and prove this gives rank bounds equivalent to those obtained from a 4-descent using visualisation techniques for <figure><img></figure>. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 451-481"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272640","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":"Applications of a Siegel-like formula of Eichler orders","authors":"Yan-Bin Li","doi":"10.1016/j.jnt.2025.04.010","DOIUrl":"10.1016/j.jnt.2025.04.010","url":null,"abstract":"<div><div>We obtain a Siegel-like formula on all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rationals. Applying it, we give three unified formulae for the number of elements with a same reduced norm and reduced trace in one of the Eichler orders in three cases respectively, when the type number of the Eichler orders equals one. If these Eichler orders belong to more than one type, using the Jacobi theta series and the Jacobi Eisenstein series in the Siegel-like formula, we construct Jacobi cusp forms which correspond to certain cusp forms of weight 3/2 in Kohnen's plus space by Gross' construction. The latter can help to provide certain newforms of weight 3/2 in Kohnen's plus space whose Shimura lift are the trace forms of certain subspaces of the space of newforms of weight 2. These trace forms are sometimes the newforms of weight 2 corresponding to elliptic curves over the rational numbers with a squarefree conductor, whose modified <em>L</em>-functions have even functional equations.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"278 ","pages":"Pages 64-111"},"PeriodicalIF":0.6,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144243505","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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