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Sato-Tate distribution of p-adic hypergeometric functions. p进超几何函数的Sato-Tate分布。
IF 0.6
Research in Number Theory Pub Date : 2023-01-01 Epub Date: 2022-11-29 DOI: 10.1007/s40993-022-00414-w
Sudhir Pujahari, Neelam Saikia
{"title":"Sato-Tate distribution of <i>p</i>-adic hypergeometric functions.","authors":"Sudhir Pujahari, Neelam Saikia","doi":"10.1007/s40993-022-00414-w","DOIUrl":"10.1007/s40993-022-00414-w","url":null,"abstract":"<p><p>Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the <i>p</i>-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the <i>p</i>-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of <i>p</i>th Hecke operators acting on the spaces of cusp forms of even weight <math><mrow><mi>k</mi> <mo>≥</mo> <mn>4</mn></mrow> </math> and levels 4 and 8 in terms of <i>p</i>-adic hypergeometric function which is of independent interest. These results can be viewed as <i>p</i>-adic analogous of some trace formulas of [1, 2, 6].</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":" ","pages":"4"},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9708806/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"35346191","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}
引用次数: 0
Summing μ ( n ) : a faster elementary algorithm. 求和μ (n):一个更快的初等算法。
IF 0.8
Research in Number Theory Pub Date : 2023-01-01 DOI: 10.1007/s40993-022-00408-8
Harald Andrés Helfgott, Lola Thompson
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Summing <ns0:math><ns0:mrow><ns0:mi>μ</ns0:mi> <ns0:mo>(</ns0:mo> <ns0:mi>n</ns0:mi> <ns0:mo>)</ns0:mo></ns0:mrow> </ns0:math> : a faster elementary algorithm.","authors":"Harald Andrés Helfgott,&nbsp;Lola Thompson","doi":"10.1007/s40993-022-00408-8","DOIUrl":"https://doi.org/10.1007/s40993-022-00408-8","url":null,"abstract":"<p><p>We present a new elementary algorithm that takes <math><mrow><mtext>time</mtext> <mspace></mspace> <mspace></mspace> <msub><mi>O</mi> <mi>ϵ</mi></msub> <mfenced><msup><mi>x</mi> <mfrac><mn>3</mn> <mn>5</mn></mfrac> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mrow><mfrac><mn>8</mn> <mn>5</mn></mfrac> <mo>+</mo> <mi>ϵ</mi></mrow> </msup> </mfenced> <mspace></mspace> <mspace></mspace> <mtext>and</mtext> <mspace></mspace> <mtext>space</mtext> <mspace></mspace> <mspace></mspace> <mi>O</mi> <mfenced><msup><mi>x</mi> <mfrac><mn>3</mn> <mn>10</mn></mfrac> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mfrac><mn>13</mn> <mn>10</mn></mfrac> </msup> </mfenced> </mrow> </math> (measured bitwise) for computing <math><mrow><mi>M</mi> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> <mo>=</mo> <msub><mo>∑</mo> <mrow><mi>n</mi> <mo>≤</mo> <mi>x</mi></mrow> </msub> <mi>μ</mi> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> <mo>,</mo></mrow> </math> where <math><mrow><mi>μ</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> is the Möbius function. This is the first improvement in the exponent of <i>x</i> for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to <math><mrow><mi>O</mi> <mo>(</mo> <msup><mi>x</mi> <mrow><mn>1</mn> <mo>/</mo> <mn>5</mn></mrow> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mrow><mn>5</mn> <mo>/</mo> <mn>3</mn></mrow> </msup> <mo>)</mo></mrow> </math> by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of <math> <mrow><msup><mi>x</mi> <mrow><mn>3</mn> <mo>/</mo> <mn>5</mn></mrow> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mn>2</mn></msup> <mo>log</mo> <mo>log</mo> <mi>x</mi></mrow> </math> .</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"9 1","pages":"6"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9731940/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10687746","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}
引用次数: 0
Geometric sieve over number fields for higher moments. 在数字域上进行几何筛选以获得更高的矩。
IF 0.8
Research in Number Theory Pub Date : 2023-01-01 Epub Date: 2023-08-02 DOI: 10.1007/s40993-023-00466-6
Giacomo Micheli, Severin Schraven, Simran Tinani, Violetta Weger
{"title":"Geometric sieve over number fields for higher moments.","authors":"Giacomo Micheli,&nbsp;Severin Schraven,&nbsp;Simran Tinani,&nbsp;Violetta Weger","doi":"10.1007/s40993-023-00466-6","DOIUrl":"10.1007/s40993-023-00466-6","url":null,"abstract":"<p><p>The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field <i>K</i>. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"9 3","pages":"62"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10397131/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9943903","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}
引用次数: 0
p-adic vertex operator algebras. p进顶点算子代数。
IF 0.8
Research in Number Theory Pub Date : 2023-01-01 DOI: 10.1007/s40993-023-00433-1
Cameron Franc, Geoffrey Mason
{"title":"<i>p</i>-adic vertex operator algebras.","authors":"Cameron Franc,&nbsp;Geoffrey Mason","doi":"10.1007/s40993-023-00433-1","DOIUrl":"https://doi.org/10.1007/s40993-023-00433-1","url":null,"abstract":"<p><p>We postulate axioms for a chiral half of a nonarchimedean 2-dimensional bosonic conformal field theory, that is, a vertex operator algebra in which a <i>p</i>-adic Banach space replaces the traditional Hilbert space. We study some consequences of our axioms leading to the construction of various examples, including <i>p</i>-adic commutative Banach rings and <i>p</i>-adic versions of the Virasoro, Heisenberg, and the Moonshine module vertex operator algebras. Serre <i>p</i>-adic modular forms occur naturally in some of these examples as limits of classical 1-point functions.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"9 2","pages":"27"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10071837/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9289144","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}
引用次数: 2
Kampé de Fériet hypergeometric functions over finite fields 有限域上的超几何函数
IF 0.8
Research in Number Theory Pub Date : 2022-12-29 DOI: 10.1007/s40993-023-00460-y
Ryojun Ito, Satoshi Kumabe, Akio Nakagawa, Yusuke Nemoto
{"title":"Kampé de Fériet hypergeometric functions over finite fields","authors":"Ryojun Ito, Satoshi Kumabe, Akio Nakagawa, Yusuke Nemoto","doi":"10.1007/s40993-023-00460-y","DOIUrl":"https://doi.org/10.1007/s40993-023-00460-y","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90279150","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}
引用次数: 0
Fast change of level and applications to isogenies 水平的快速变化和应用到同基因
IF 0.8
Research in Number Theory Pub Date : 2022-12-12 DOI: 10.1007/s40993-022-00407-9
D. Lubicz, Damien Robert
{"title":"Fast change of level and applications to isogenies","authors":"D. Lubicz, Damien Robert","doi":"10.1007/s40993-022-00407-9","DOIUrl":"https://doi.org/10.1007/s40993-022-00407-9","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81124136","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}
引用次数: 12
Triangular modular curves of small genus 小属的三角形模曲线
IF 0.8
Research in Number Theory Pub Date : 2022-11-29 DOI: 10.1007/s40993-022-00395-w
Juanita Duque-Rosero, J. Voight
{"title":"Triangular modular curves of small genus","authors":"Juanita Duque-Rosero, J. Voight","doi":"10.1007/s40993-022-00395-w","DOIUrl":"https://doi.org/10.1007/s40993-022-00395-w","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"285 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72443235","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}
引用次数: 0
Rogers–Ramanujan type generalized reciprocal identities and Eisenstein series Rogers-Ramanujan型广义互反恒等式与爱森斯坦级数
IF 0.8
Research in Number Theory Pub Date : 2022-11-29 DOI: 10.1007/s40993-022-00416-8
Zafer Selcuk Aygin
{"title":"Rogers–Ramanujan type generalized reciprocal identities and Eisenstein series","authors":"Zafer Selcuk Aygin","doi":"10.1007/s40993-022-00416-8","DOIUrl":"https://doi.org/10.1007/s40993-022-00416-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88716266","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}
引用次数: 0
Modular forms via invariant theory 模形式通过不变理论
IF 0.8
Research in Number Theory Pub Date : 2022-11-10 DOI: 10.1007/s40993-023-00442-0
Fabien Cl'ery, G. van der Geer
{"title":"Modular forms via invariant theory","authors":"Fabien Cl'ery, G. van der Geer","doi":"10.1007/s40993-023-00442-0","DOIUrl":"https://doi.org/10.1007/s40993-023-00442-0","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"47 1","pages":"1-10"},"PeriodicalIF":0.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80664305","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}
引用次数: 1
Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3 3属超椭圆曲线雅可比矩阵扭转子群的计算
IF 0.8
Research in Number Theory Pub Date : 2022-11-07 DOI: 10.1007/s40993-023-00429-x
J. Müller, Berno Reitsma
{"title":"Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3","authors":"J. Müller, Berno Reitsma","doi":"10.1007/s40993-023-00429-x","DOIUrl":"https://doi.org/10.1007/s40993-023-00429-x","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80787292","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}
引用次数: 1
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