Pub Date : 2023-10-13DOI: 10.1142/s1793042124500210
Xin Wang, Tengyou Zhu
Let $pi$ be a $SL(3,mathbb Z)$ Hecke-Maass cusp form and $chi$ a primitive Dirichlet character of prime power conductor $mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ Lleft(frac{1}{2}+it,pitimes chiright)ll_{pi,varepsilon} p^{3/4}big(mathfrak{q}(1+|t|)big)^{3/4-3/40+varepsilon}, $$ for any $varepsilon>0$ and $t in mathbb{R}$.
{"title":"Hybrid Subconvexity Bounds for Twists of GL(3) <i>L</i>-Functions","authors":"Xin Wang, Tengyou Zhu","doi":"10.1142/s1793042124500210","DOIUrl":"https://doi.org/10.1142/s1793042124500210","url":null,"abstract":"Let $pi$ be a $SL(3,mathbb Z)$ Hecke-Maass cusp form and $chi$ a primitive Dirichlet character of prime power conductor $mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ Lleft(frac{1}{2}+it,pitimes chiright)ll_{pi,varepsilon} p^{3/4}big(mathfrak{q}(1+|t|)big)^{3/4-3/40+varepsilon}, $$ for any $varepsilon>0$ and $t in mathbb{R}$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135917870","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/s1793042124500416
Mahesh Kumar Ram
{"title":"Determination of all Imaginary Cyclic Quartic Fields of Prime Class Number p ≡ 3(mod4), and non-divisibility of class numbers","authors":"Mahesh Kumar Ram","doi":"10.1142/s1793042124500416","DOIUrl":"https://doi.org/10.1142/s1793042124500416","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918184","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/s179304212450026x
Aritram Dhar
A BSTRACT . In this paper, we provide proofs of two 5 ψ 5 summation formulas of Bailey using a 5 φ 4 identity of Carlitz. We show that in the limiting case, the two 5 ψ 5 identities give rise to two 3 ψ 3 summation formulas of Bailey. Finally, we prove the two 3 ψ 3 identities using a technique initially used by Ismail to prove Ramanujan’s 1 ψ 1 summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised 6 ψ 6 sum.
{"title":"On 5ψ5 Identities of Bailey","authors":"Aritram Dhar","doi":"10.1142/s179304212450026x","DOIUrl":"https://doi.org/10.1142/s179304212450026x","url":null,"abstract":"A BSTRACT . In this paper, we provide proofs of two 5 ψ 5 summation formulas of Bailey using a 5 φ 4 identity of Carlitz. We show that in the limiting case, the two 5 ψ 5 identities give rise to two 3 ψ 3 summation formulas of Bailey. Finally, we prove the two 3 ψ 3 identities using a technique initially used by Ismail to prove Ramanujan’s 1 ψ 1 summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised 6 ψ 6 sum.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918478","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/s1793042124500295
Pramath Anamby
We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_mu$ with either $(mu,2m_1m_2)=1$ or $(mu,2m_1m_2)nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.
{"title":"Non-Vanishing of theta Components of Jacobi Forms with Level and an Application","authors":"Pramath Anamby","doi":"10.1142/s1793042124500295","DOIUrl":"https://doi.org/10.1142/s1793042124500295","url":null,"abstract":"We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_mu$ with either $(mu,2m_1m_2)=1$ or $(mu,2m_1m_2)nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918622","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/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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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