Pub Date : 2023-10-13DOI: 10.1142/s1793042124500453
Angel Kumchev, Nathan Mcnew, Ariana Park
Let $k geq 2$ be an integer and $mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $mathbb F_q[x]$ that are not divisible by the $k$th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all $k ge 2$. We also develop polynomial versions of the classical techniques used to study gaps between $k$-free integers in $mathbb Z$. We apply these techniques to obtain analogues in $mathbb F_q[x]$ of some classical theorems on the distribution of $k$-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.
{"title":"Short Interval Results for Powerfree Polynomials Over Finite Fields","authors":"Angel Kumchev, Nathan Mcnew, Ariana Park","doi":"10.1142/s1793042124500453","DOIUrl":"https://doi.org/10.1142/s1793042124500453","url":null,"abstract":"Let $k geq 2$ be an integer and $mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $mathbb F_q[x]$ that are not divisible by the $k$th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all $k ge 2$. We also develop polynomial versions of the classical techniques used to study gaps between $k$-free integers in $mathbb Z$. We apply these techniques to obtain analogues in $mathbb F_q[x]$ of some classical theorems on the distribution of $k$-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918465","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500398
Sara Arias-de-Reyna, Joachim Konig
Using Galois representations attached to elliptic curves, we construct Galois extensions of $mathbb{Q}$ with group $GL_2(p)$ in which all decomposition groups are cyclic. This is the first such realization for all primes $p$.
{"title":"Locally cyclic extensions with Galois group GL<sub>2</sub>(<i>p</i>)","authors":"Sara Arias-de-Reyna, Joachim Konig","doi":"10.1142/s1793042124500398","DOIUrl":"https://doi.org/10.1142/s1793042124500398","url":null,"abstract":"Using Galois representations attached to elliptic curves, we construct Galois extensions of $mathbb{Q}$ with group $GL_2(p)$ in which all decomposition groups are cyclic. This is the first such realization for all primes $p$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918469","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500349
Ilija Vrecica
{"title":"A Note on the size of Iterated Sumsets in ℤ<sup><i>d</i></sup>","authors":"Ilija Vrecica","doi":"10.1142/s1793042124500349","DOIUrl":"https://doi.org/10.1142/s1793042124500349","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918470","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500234
Jonah Klein, Dimitris Koukoulopoulos, Simon Lemieux
A BSTRACT . Covering systems were introduced by Erd˝os in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister, Bollob´as, Morris, Sahasrabudhe and Tiba gave a simpler and more versatile proof of Hough’s result. Building upon their work, we show that there exists some absolute constant c > 0 such that the j -th smallest modulus of a minimal covering system with distinct moduli is (cid:54) exp( cj 2 / log( j + 1)) .
{"title":"On The <i>j</i>-TH Smallest Modulus of a Covering System with Distinct Moduli","authors":"Jonah Klein, Dimitris Koukoulopoulos, Simon Lemieux","doi":"10.1142/s1793042124500234","DOIUrl":"https://doi.org/10.1142/s1793042124500234","url":null,"abstract":"A BSTRACT . Covering systems were introduced by Erd˝os in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister, Bollob´as, Morris, Sahasrabudhe and Tiba gave a simpler and more versatile proof of Hough’s result. Building upon their work, we show that there exists some absolute constant c > 0 such that the j -th smallest modulus of a minimal covering system with distinct moduli is (cid:54) exp( cj 2 / log( j + 1)) .","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918624","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500301
Sam Frengley
. We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over Q with 12-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen [Che16] and Fisher [Fis20] where it is assumed that the underlying isomorphism of 12-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the 2, 3, or 4-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j -invariants of the pair of elliptic curves.
{"title":"On 12-Congruences of Elliptic Curves","authors":"Sam Frengley","doi":"10.1142/s1793042124500301","DOIUrl":"https://doi.org/10.1142/s1793042124500301","url":null,"abstract":". We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over Q with 12-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen [Che16] and Fisher [Fis20] where it is assumed that the underlying isomorphism of 12-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the 2, 3, or 4-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j -invariants of the pair of elliptic curves.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918039","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500441
Sheng-Chi Liu, Jakob Streipel
{"title":"The Twisted Second Moment of <i>L</i>-Functions Associated to Hecke-Maass Forms","authors":"Sheng-Chi Liu, Jakob Streipel","doi":"10.1142/s1793042124500441","DOIUrl":"https://doi.org/10.1142/s1793042124500441","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918463","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500313
Yue-Feng She, Hai-Liang Wu
In this paper, we study several topics on additive decompositions of primitive elemements in finite fields. Also we refine some bounds obtained by Dartyge and S'{a}rk"{o}zy and Shparlinski.
{"title":"On Additive Decompositions of Primitive Elements in Finite Fields","authors":"Yue-Feng She, Hai-Liang Wu","doi":"10.1142/s1793042124500313","DOIUrl":"https://doi.org/10.1142/s1793042124500313","url":null,"abstract":"In this paper, we study several topics on additive decompositions of primitive elemements in finite fields. Also we refine some bounds obtained by Dartyge and S'{a}rk\"{o}zy and Shparlinski.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918623","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500404
Fei Hou, Guangshi Lu
{"title":"An explicit Voronoi Formula for <i>SL</i><sub>3</sub>(ℝ) Newforms Underlying the Symmetric Lifts in the Level Aspect","authors":"Fei Hou, Guangshi Lu","doi":"10.1142/s1793042124500404","DOIUrl":"https://doi.org/10.1142/s1793042124500404","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918626","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500374
Abbas Maarefparvar
{"title":"The Ostrowski Quotient of an Elliptic Curve","authors":"Abbas Maarefparvar","doi":"10.1142/s1793042124500374","DOIUrl":"https://doi.org/10.1142/s1793042124500374","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918924","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}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500271
Yuqing Zhang
{"title":"Metrical Properties for Functions of Consecutive Multiple Partial Quotients in Continued Fractions","authors":"Yuqing Zhang","doi":"10.1142/s1793042124500271","DOIUrl":"https://doi.org/10.1142/s1793042124500271","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918149","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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