Pub Date : 2023-10-13DOI: 10.1142/s1793042124500350
Tarun Dalal
In this article, we determine all intermediate modular curves $X_Delta(N)$ that admit infinitely many cubic points over the rational field $mathbb{Q}$.
{"title":"Intermediate modular curves with infinitely many cubic points over ℚ","authors":"Tarun Dalal","doi":"10.1142/s1793042124500350","DOIUrl":"https://doi.org/10.1142/s1793042124500350","url":null,"abstract":"In this article, we determine all intermediate modular curves $X_Delta(N)$ that admit infinitely many cubic points over the rational field $mathbb{Q}$.","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":"135918183","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/s1793042124500325
Hirotaka Kobayashi
{"title":"On some discrete mean values of higher derivatives of Hardy's <i>Z</i>-Function","authors":"Hirotaka Kobayashi","doi":"10.1142/s1793042124500325","DOIUrl":"https://doi.org/10.1142/s1793042124500325","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"32 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":"135918920","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/s1793042124500337
Tomoaki Nakaya
The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from $q=e^{2pi i tau}$ to the reciprocal of the elliptic modular $j$-function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of $p$-adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them.
{"title":"Determination of normalized extremal quasimodular forms of depth 1 with integral Fourier coefficients","authors":"Tomoaki Nakaya","doi":"10.1142/s1793042124500337","DOIUrl":"https://doi.org/10.1142/s1793042124500337","url":null,"abstract":"The main purpose of this paper is to determine all normalized extremal quasimodular forms of depth 1 whose Fourier coefficients are integers. By changing the local parameter at infinity from $q=e^{2pi i tau}$ to the reciprocal of the elliptic modular $j$-function, we prove that all normalized extremal quasimodular forms of depth 1 have a hypergeometric series expression and that integrality is not affected by this change of parameters. Furthermore, by transforming these hypergeometric series expressions into a certain manageable form related to the Atkin(-like) polynomials and using the lemmas that appeared in the study of $p$-adic hypergeometric series by Dwork and Zudilin, the integrality problem can be reduced to the fact that a polynomial vanishes modulo a prime power, which we prove. We also prove that all extremal quasimodular forms of depth 1 with appropriate weight-dependent leading coefficients have integral Fourier coefficients by focusing on the hypergeometric expression of them.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"28 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":"135918015","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/s1793042124500258
Carlos A. Gomez, Salah Eddine Rihane, Alain Togbe
{"title":"On The <i>X</i>-Coordinates of Pell Equations X2 − dY2 = ±1 As Difference of two Fibonacci Numbers","authors":"Carlos A. Gomez, Salah Eddine Rihane, Alain Togbe","doi":"10.1142/s1793042124500258","DOIUrl":"https://doi.org/10.1142/s1793042124500258","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"69 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":"135918483","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/s1793042124500428
Mikhail A. Reynya
{"title":"A Two-Parametric Family of High Rank Mordell Curves","authors":"Mikhail A. Reynya","doi":"10.1142/s1793042124500428","DOIUrl":"https://doi.org/10.1142/s1793042124500428","url":null,"abstract":"","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":"135918168","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/s1793042124500167
Deepesh Singhal, Yuxin Lin
Denote the set of algebraic numbers as $overline{mathbb{Q}}$ and the set of algebraic integers as $overline{mathbb{Z}}$. For $gammainoverline{mathbb{Q}}$, consider its irreducible polynomial in $mathbb{Z}[x]$, $F_{gamma}(x)=a_nx^n+dots+a_0$. Denote $e(gamma)=gcd(a_{n},a_{n-1},dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $mathbb{Z}[gamma]cap mathbb{Q}={alphainmathbb{Q}mid {pmid v_p(alpha)<0}subseteq {pmid p|e(gamma)}}$. Given a number field $K$ and $gammainoverline{mathbb{Q}}$, we show that there is a subset $X(K,gamma)subseteq text{Spec}(mathcal{O}_K)$, for which $mathcal{O}_K[gamma]cap K={alphain Kmid {mathfrak{p}mid v_{mathfrak{p}}(alpha)<0}subseteq X(K,gamma)}$. We prove that $mathcal{O}_K[gamma]cap K$ is a principal ideal domain if and only if the primes in $X(K,gamma)$ generate the class group of $mathcal{O}_K$. We show that given $gammain overline{mathbb{Q}}$, we can find a finite set $Ssubseteq overline{mathbb{Z}}$, such that for every number field $K$, we have $X(K,gamma)={mathfrak{p}intext{Spec}(mathcal{O}_K)mid mathfrak{p}cap Sneq emptyset}$. We study how this set $S$ relates to the ring $overline{mathbb{Z}}[gamma]$ and the ideal $mathfrak{D}_{gamma}={ainoverline{mathbb{Z}}mid agammainoverline{mathbb{Z}}}$ of $overline{mathbb{Z}}$. We also show that $gamma_1,gamma_2in overline{mathbb{Q}}$ satisfy $mathfrak{D}_{gamma_1}=mathfrak{D}_{gamma_2}$ if and only if $X(K,gamma_1)=X(K,gamma_2)$ for all number fields $K$.
{"title":"Primes in denominators of algebraic numbers","authors":"Deepesh Singhal, Yuxin Lin","doi":"10.1142/s1793042124500167","DOIUrl":"https://doi.org/10.1142/s1793042124500167","url":null,"abstract":"Denote the set of algebraic numbers as $overline{mathbb{Q}}$ and the set of algebraic integers as $overline{mathbb{Z}}$. For $gammainoverline{mathbb{Q}}$, consider its irreducible polynomial in $mathbb{Z}[x]$, $F_{gamma}(x)=a_nx^n+dots+a_0$. Denote $e(gamma)=gcd(a_{n},a_{n-1},dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $mathbb{Z}[gamma]cap mathbb{Q}={alphainmathbb{Q}mid {pmid v_p(alpha)<0}subseteq {pmid p|e(gamma)}}$. Given a number field $K$ and $gammainoverline{mathbb{Q}}$, we show that there is a subset $X(K,gamma)subseteq text{Spec}(mathcal{O}_K)$, for which $mathcal{O}_K[gamma]cap K={alphain Kmid {mathfrak{p}mid v_{mathfrak{p}}(alpha)<0}subseteq X(K,gamma)}$. We prove that $mathcal{O}_K[gamma]cap K$ is a principal ideal domain if and only if the primes in $X(K,gamma)$ generate the class group of $mathcal{O}_K$. We show that given $gammain overline{mathbb{Q}}$, we can find a finite set $Ssubseteq overline{mathbb{Z}}$, such that for every number field $K$, we have $X(K,gamma)={mathfrak{p}intext{Spec}(mathcal{O}_K)mid mathfrak{p}cap Sneq emptyset}$. We study how this set $S$ relates to the ring $overline{mathbb{Z}}[gamma]$ and the ideal $mathfrak{D}_{gamma}={ainoverline{mathbb{Z}}mid agammainoverline{mathbb{Z}}}$ of $overline{mathbb{Z}}$. We also show that $gamma_1,gamma_2in overline{mathbb{Q}}$ satisfy $mathfrak{D}_{gamma_1}=mathfrak{D}_{gamma_2}$ if and only if $X(K,gamma_1)=X(K,gamma_2)$ for all number fields $K$.","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":"135918316","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/s1793042124500192
Bernhard Heim, Markus Neuhauser
We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${g_d(n)}_{dgeq 0,n geq 1}$ be the double sequences $sigma_d(n)= sum_{ell mid n} ell^d$ or $psi_d(n)= n^d$. We associate double sequences $left{ p^{g_{d} }left( nright) right}$ and $left{ q^{g_{d} }left( nright) right} $, defined as the coefficients of begin{eqnarray*} sum_{n=0}^{infty} p^{g_{d} }left( nright) , t^{n}&:=&prod_{n=1}^{infty} left( 1 - t^{n} right)^{-frac{ sum_{ell mid n} mu(ell) , g_d(n/ell) }{n} }, sum_{n=0}^{infty} q^{g_{d} }left( nright) , t^{n}&:=&frac{1}{1 - sum_{n=1}^{infty} g_d(n) , t^{n} }. end{eqnarray*} These coefficients are related to the number of partitions $mathrm{p}left( nright) = p^{sigma _{1 }}left ( nright) $, plane partitions $ppleft( nright) = p^{sigma _{2 }}left( nright) $ of $n$, and Fibonacci numbers $F_{2n} = q^{psi _{1 }}left( nright) $. Let $n geq 3$ and let $n equiv 0 pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n equiv 2 pmod{3}$. Let $nequiv 1 pmod{3}$. Then the log-concave property flips for almost all $d$.
{"title":"Log-Concavity of Infinite Product and Infinite Sum Generating Functions","authors":"Bernhard Heim, Markus Neuhauser","doi":"10.1142/s1793042124500192","DOIUrl":"https://doi.org/10.1142/s1793042124500192","url":null,"abstract":"We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${g_d(n)}_{dgeq 0,n geq 1}$ be the double sequences $sigma_d(n)= sum_{ell mid n} ell^d$ or $psi_d(n)= n^d$. We associate double sequences $left{ p^{g_{d} }left( nright) right}$ and $left{ q^{g_{d} }left( nright) right} $, defined as the coefficients of begin{eqnarray*} sum_{n=0}^{infty} p^{g_{d} }left( nright) , t^{n}&:=&prod_{n=1}^{infty} left( 1 - t^{n} right)^{-frac{ sum_{ell mid n} mu(ell) , g_d(n/ell) }{n} }, sum_{n=0}^{infty} q^{g_{d} }left( nright) , t^{n}&:=&frac{1}{1 - sum_{n=1}^{infty} g_d(n) , t^{n} }. end{eqnarray*} These coefficients are related to the number of partitions $mathrm{p}left( nright) = p^{sigma _{1 }}left ( nright) $, plane partitions $ppleft( nright) = p^{sigma _{2 }}left( nright) $ of $n$, and Fibonacci numbers $F_{2n} = q^{psi _{1 }}left( nright) $. Let $n geq 3$ and let $n equiv 0 pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n equiv 2 pmod{3}$. Let $nequiv 1 pmod{3}$. Then the log-concave property flips for almost all $d$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"37 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":"135918468","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/s1793042124500386
Byoung Du Kim
{"title":"Iwasawa theory of plus/minus Selmer groups with non-co-free plus/minus local conditions","authors":"Byoung Du Kim","doi":"10.1142/s1793042124500386","DOIUrl":"https://doi.org/10.1142/s1793042124500386","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":"135918474","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/s1793042124500222
William Frendreiss, Jennifer Gao, Austin Lei, Amy Woodall, Hui Xue, Daozhou Zhu
{"title":"Interlacing Properties for Zeros of a Family of Modular Forms","authors":"William Frendreiss, Jennifer Gao, Austin Lei, Amy Woodall, Hui Xue, Daozhou Zhu","doi":"10.1142/s1793042124500222","DOIUrl":"https://doi.org/10.1142/s1793042124500222","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"13 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":"135918482","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 : 2023-08-04DOI: 10.1142/s1793042124500088
Xuejun Guo, Q. Ji, Hang Liu, H. Qin
{"title":"The mahler measure of x + 1/x + y + 1/y + 4 ± 42 and Beilinson’s conjecture","authors":"Xuejun Guo, Q. Ji, Hang Liu, H. Qin","doi":"10.1142/s1793042124500088","DOIUrl":"https://doi.org/10.1142/s1793042124500088","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82963127","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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