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Acta Arithmetica最新文献

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On Mahler’s inequality and small integral generators of totally complex number fields 论马勒不等式和完全复数域的小积分生成器
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-08-09 DOI: 10.4064/aa230601-18-9
Murray Child, Martin Widmer
{"title":"On Mahler’s inequality and small integral generators of totally complex number fields","authors":"Murray Child, Martin Widmer","doi":"10.4064/aa230601-18-9","DOIUrl":"https://doi.org/10.4064/aa230601-18-9","url":null,"abstract":"We improve Mahler's lower bound for the Mahler measure in terms of the discriminant and degree for a specific class of polynomials: complex monic polynomials of degree $dgeq 2$ such that all roots with modulus greater than some fixed value $rgeq1$ occur in equal modulus pairs. We improve Mahler's exponent $frac{1}{2d-2}$ on the discriminant to $frac{1}{2d-3}$. Moreover, we show that this value is sharp, even when restricting to minimal polynomials of integral generators of a fixed not totally real number field. An immediate consequence of this new lower bound is an improved lower bound for integral generators of number fields, generalising a simple observation of Ruppert from imaginary quadratic to totally complex number fields of arbitrary degree.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"70 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139351233","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}
引用次数: 0
On a simple quartic family of Thue equations over imaginary quadratic number fields 虚二次数域上的一个简单四次族Thue方程
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-03-27 DOI: 10.4064/aa230329-19-6
B. Earp-Lynch, Bernadette Faye, E. Goedhart, I. Vukusic, Daniel P. Wisniewski
{"title":"On a simple quartic family of Thue equations over imaginary quadratic number fields","authors":"B. Earp-Lynch, Bernadette Faye, E. Goedhart, I. Vukusic, Daniel P. Wisniewski","doi":"10.4064/aa230329-19-6","DOIUrl":"https://doi.org/10.4064/aa230329-19-6","url":null,"abstract":"Let $t$ be any imaginary quadratic integer with $|t|geq 100$. We prove that the inequality [ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | leq 1 ] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| leq C|t|$ and $|F_t(X,Y)| leq |t|^{2 -varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44168848","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}
引用次数: 0
Ultra-short sums of trace functions 跟踪函数的超短和
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-02-27 DOI: 10.4064/aa230308-11-5
E. Kowalski, Th'eo Untrau
{"title":"Ultra-short sums of trace functions","authors":"E. Kowalski, Th'eo Untrau","doi":"10.4064/aa230308-11-5","DOIUrl":"https://doi.org/10.4064/aa230308-11-5","url":null,"abstract":"We generalize results of Duke, Garcia, Hyde, Lutz and others on the distribution of sums of roots of unity related to Gaussian periods to obtain equidistribution of similar sums over zeros of arbitrary integral polynomials. We also interpret these results in terms of trace functions, and generalize them to higher rank trace functions.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41947090","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}
引用次数: 0
Growth of $p$-parts of ideal class groups and fine Selmer groups in $mathbb Z_q$-extensions with $pne q$ $mathbb Z_q$中理想类群和精细Selmer群的$p$-部分的增长- $pne q$的扩展
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-02-27 DOI: 10.4064/aa220518-28-2
Debanjana Kundu, Antonio Lei
{"title":"Growth of $p$-parts of ideal class groups and fine Selmer groups in $mathbb Z_q$-extensions with $pne q$","authors":"Debanjana Kundu, Antonio Lei","doi":"10.4064/aa220518-28-2","DOIUrl":"https://doi.org/10.4064/aa220518-28-2","url":null,"abstract":"Fix two distinct odd primes $p$ and $q$. We study\"$pne q$\"Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $mathbb{Z}_q$-extension of $K$. Let $F$ be a number field and let $A_{/F}$ be an abelian variety with $A[p]subseteq A(F)$. We give sufficient conditions for the $p$-part of the fine Selmer groups of $A$ over finite subextensions of a $mathbb{Z}_q$-extension of $F$ to stabilize.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45647920","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}
引用次数: 0
On the structure of even $K$-groups of rings of algebraic integers 代数整数环的偶K群的结构
3区 数学
Acta Arithmetica Pub Date : 2023-01-01 DOI: 10.4064/aa221029-25-7
Meng Fai Lim
{"title":"On the structure of even $K$-groups of rings of algebraic integers","authors":"Meng Fai Lim","doi":"10.4064/aa221029-25-7","DOIUrl":"https://doi.org/10.4064/aa221029-25-7","url":null,"abstract":"We describe the higher even $K$-groups of the ring of integers of a number field in terms of the class groups of an appropriate extension of the number field in question. This is a natural extension of the previous work of Browkin, Keune and Kolster, who","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135051529","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}
引用次数: 0
The irrationality of a divisor function series of Erdős and Kac Erdős和Kac的一个除数函数级数的无理性
3区 数学
Acta Arithmetica Pub Date : 2023-01-01 DOI: 10.4064/aa220927-1-9
Kyle Pratt
{"title":"The irrationality of a divisor function series of Erdős and Kac","authors":"Kyle Pratt","doi":"10.4064/aa220927-1-9","DOIUrl":"https://doi.org/10.4064/aa220927-1-9","url":null,"abstract":"For positive integers $k$ and $n$ let $sigma _k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number $alpha _k = sum _{ngeq 1} frac {sigma _k(n)}{n!}$ is irrational. It is known uncond","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135559808","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}
引用次数: 0
Lower bounds for Rankin–Selberg $L$-functions on the edge of the critical strip 临界带边缘上Rankin-Selberg $L$-函数的下界
3区 数学
Acta Arithmetica Pub Date : 2023-01-01 DOI: 10.4064/aa221111-14-7
Qiao Zhang
{"title":"Lower bounds for Rankin–Selberg $L$-functions on the edge of the critical strip","authors":"Qiao Zhang","doi":"10.4064/aa221111-14-7","DOIUrl":"https://doi.org/10.4064/aa221111-14-7","url":null,"abstract":"Let $F$ be a number field, and let $pi_1$ and $pi_2$ be distinct unitary cuspidal automorphic representations of $operatorname{GL}_{n_1}(mathbb{A}_F)$ and $operatorname{GL}_{n_2}(mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, pi_1 times widetilde{pi}_2)$ along the edge $Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, pi_1 times widetilde{pi}_2)$ is also determined.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136260001","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}
引用次数: 0
Ramanujan–Sato series for $1/pi $ $1/pi $的Ramanujan-Sato级数
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-01-01 DOI: 10.4064/aa220621-19-12
Tim Huber, Daniel Schultz, Dongxi Ye
{"title":"Ramanujan–Sato series for $1/pi $","authors":"Tim Huber, Daniel Schultz, Dongxi Ye","doi":"10.4064/aa220621-19-12","DOIUrl":"https://doi.org/10.4064/aa220621-19-12","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440175","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}
引用次数: 1
Diophantine equations for Littlewood polynomials 利特伍德多项式的丢番图方程
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-01-01 DOI: 10.4064/aa220912-3-11
L. Hajdu, R. Tijdeman, N. Varga
{"title":"Diophantine equations for Littlewood polynomials","authors":"L. Hajdu, R. Tijdeman, N. Varga","doi":"10.4064/aa220912-3-11","DOIUrl":"https://doi.org/10.4064/aa220912-3-11","url":null,"abstract":". In this paper we give finiteness results for the shifted power values and polynomial values of Littlewood polynomials.","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440351","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}
引用次数: 0
Additive decomposition of signed primes 有符号素数的加性分解
IF 0.7 3区 数学
Acta Arithmetica Pub Date : 2023-01-01 DOI: 10.4064/aa220429-17-11
I. Ruzsa
{"title":"Additive decomposition of signed primes","authors":"I. Ruzsa","doi":"10.4064/aa220429-17-11","DOIUrl":"https://doi.org/10.4064/aa220429-17-11","url":null,"abstract":"","PeriodicalId":37888,"journal":{"name":"Acta Arithmetica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70440088","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}
引用次数: 1
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