Research in Number Theory最新文献

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">On <i>p</i>-refined Friedberg-Jacquet integrals and the classical symplectic locus in the <ns0:math> <ns0:msub><ns0:mrow><ns0:mspace /> <ns0:mtext>GL</ns0:mtext> <ns0:mspace /></ns0:mrow> <ns0:mrow><ns0:mn>2</ns0:mn> <ns0:mi>n</ns0:mi></ns0:mrow> </ns0:msub> </ns0:math> eigenvariety.","authors":"Daniel Barrera Salazar, Andrew Graham, Chris Williams","doi":"10.1007/s40993-025-00631-z","DOIUrl":"10.1007/s40993-025-00631-z","url":null,"abstract":"<p><p>Friedberg-Jacquet proved that if <math><mi>π</mi></math> is a cuspidal automorphic representation of <math> <mrow><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> <mrow><mo>(</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> , then <math><mi>π</mi></math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a global zeta integral <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> over <math><mrow><mi>H</mi> <mo>=</mo> <msub><mtext>GL</mtext> <mi>n</mi></msub> <mo>×</mo> <msub><mtext>GL</mtext> <mi>n</mi></msub> </mrow> </math> is non-vanishing on <math><mi>π</mi></math> . We conjecture a <i>p</i>-refined analogue: that any <i>P</i>-parahoric <i>p</i>-refinement <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a <i>P</i>-twisted version of <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> is non-vanishing on the <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> -eigenspace in <math><mi>π</mi></math> . This twisted <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> appears in all constructions of <i>p</i>-adic <i>L</i>-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the <math><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> </math> eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"11 2","pages":"51"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031854/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144028936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Three-torsion subgroups and wild conductor exponents of plane quartics.","authors":"Elvira Lupoian, James Rawson","doi":"10.1007/s40993-025-00672-4","DOIUrl":"https://doi.org/10.1007/s40993-025-00672-4","url":null,"abstract":"<p><p>In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve with a marked rational point. We describe <math><mrow><mn>3</mn> <mo>-</mo></mrow> </math> torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics. We compute the points of this zero-dimensional, degree 728 scheme first by approximation, using homotopy continuation and Newton-Raphson, and then using continued fractions to obtain accurate expressions for these points. We describe how the Galois structure of the field of definition of the 3-torsion subgroup can be used to compute local wild conductor exponents, including at <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn></mrow> </math> .</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"11 4","pages":"92"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12496290/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145240026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distribution of Andrews’ singular overpartitions \u0000 \u0000 \u0000 \u0000 $${overline{C}}_{p,1}(n)$$\u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 ¯\u0000 \u0000 ","authors":"Chiranjit Ray","doi":"10.1007/s40993-023-00496-0","DOIUrl":"https://doi.org/10.1007/s40993-023-00496-0","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"59 9","pages":"1-8"},"PeriodicalIF":0.8,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139385616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Asymptotics of commuting <ns0:math><ns0:mi>ℓ</ns0:mi></ns0:math> -tuples in symmetric groups and log-concavity.","authors":"Kathrin Bringmann, Johann Franke, Bernhard Heim","doi":"10.1007/s40993-024-00562-1","DOIUrl":"10.1007/s40993-024-00562-1","url":null,"abstract":"<p><p>Denote by <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> the number of <math><mi>ℓ</mi></math> -tuples of elements in the symmetric group <math><msub><mi>S</mi> <mi>n</mi></msub> </math> with commuting components, normalized by the order of <math><msub><mi>S</mi> <mi>n</mi></msub> </math> . In this paper, we prove asymptotic formulas for <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> . In addition, general criteria for log-concavity are shown, which can be applied to <math> <mrow><msub><mi>N</mi> <mi>ℓ</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form <math><mrow><mi>c</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> <mi>c</mi> <mo>(</mo> <mi>b</mi> <mo>)</mo> <mo>></mo> <mi>c</mi> <mo>(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>)</mo></mrow> </math> for certain families of sequences <i>c</i>(<i>n</i>).</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 4","pages":"83"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11449981/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142381962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Log concavity for unimodal sequences.","authors":"Walter Bridges, Kathrin Bringmann","doi":"10.1007/s40993-023-00490-6","DOIUrl":"10.1007/s40993-023-00490-6","url":null,"abstract":"<p><p>In this paper, we prove that the number of unimodal sequences of size <i>n</i> is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 1","pages":"6"},"PeriodicalIF":0.8,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10730647/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138832121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Pillai's Problem involving Lucas sequences of the second kind.","authors":"Sebastian Heintze, Volker Ziegler","doi":"10.1007/s40993-024-00534-5","DOIUrl":"10.1007/s40993-024-00534-5","url":null,"abstract":"<p><p>In this paper, we consider the Diophantine equation <math><mrow><msub><mi>V</mi><mi>n</mi></msub><mo>-</mo><msup><mi>b</mi><mi>m</mi></msup><mo>=</mo><mi>c</mi></mrow></math> for given integers <i>b</i>, <i>c</i> with <math><mrow><mi>b</mi><mo>≥</mo><mn>2</mn></mrow></math>, whereas <math><msub><mi>V</mi><mi>n</mi></msub></math> varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (<i>n</i>, <i>m</i>) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of <math><msub><mi>V</mi><mi>n</mi></msub></math>.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 2","pages":"51"},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11090840/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140945180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Odd degree isolated points on $$X_1(N)$$ with rational j-invariant","authors":"Abbey Bourdon, David R. Gill, Jeremy Rouse, Lori D. Watson","doi":"10.1007/s40993-023-00488-0","DOIUrl":"https://doi.org/10.1007/s40993-023-00488-0","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"347 2‐3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138966733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Error approximation for backwards and simple continued fractions","authors":"Cameron Bjorklund, Matthew Litman","doi":"10.1007/s40993-023-00481-7","DOIUrl":"https://doi.org/10.1007/s40993-023-00481-7","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"55 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139248580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transcendence criterion with $$(beta ,{mathcal {A}})$$-representations in some quadratic integer bases","authors":"Maryam Elaoud, Mohamed Hbaib","doi":"10.1007/s40993-023-00486-2","DOIUrl":"https://doi.org/10.1007/s40993-023-00486-2","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"7 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135142137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fast norm computation in smooth-degree Abelian number fields","authors":"Daniel J. Bernstein","doi":"10.1007/s40993-022-00402-0","DOIUrl":"https://doi.org/10.1007/s40993-022-00402-0","url":null,"abstract":"Abstract This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in S -unit searches (for, e.g., class-group computation) is computing a $$Theta (nlog n)$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;Θ&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; -bit norm of an element of weight $$n^{1/2+o(1)}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mi&gt;o&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;/mml:math&gt; in a degree- n field; this method then uses $$n(log n)^{3+o(1)}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:msup&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mi&gt;o&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; bit operations. An $$n(log n)^{O(1)}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:msup&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least $$n^2(log n)^{2+o(1)}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:msup&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mi&gt;o&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm $","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"116 23","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}