Refining a result of Erdős and Mays, we give asymptotic series expansions for the functions A(x)-C(x), the count of n≤x for which every group of order n is abelian (but not all cyclic), and N(x)-A(x), the count of n≤x for which every group of order n is nilpotent (but not all abelian).
{"title":"Numbers which are only orders of abelian or nilpotent groups","authors":"Matthew Just","doi":"10.5802/jtnb.1251","DOIUrl":"https://doi.org/10.5802/jtnb.1251","url":null,"abstract":"Refining a result of Erdős and Mays, we give asymptotic series expansions for the functions A(x)-C(x), the count of n≤x for which every group of order n is abelian (but not all cyclic), and N(x)-A(x), the count of n≤x for which every group of order n is nilpotent (but not all abelian).","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The continuous version of a famous result of Khinchin says that a half-infinite torus line in the unit square [0,1] 2 exhibits superdensity, a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. We extend this result of Khinchin to the case when the unit torus [0,1] 2 is replaced by a finite polysquare translation surface, or square tiled surface. In particular, we show that it is possible to study this very number-theoretic problem by restricting to traditional tools in number theory, using only continued fractions and the famous 3-distance theorem in diophantine approximation combined with an iterative process.
{"title":"Generalization of a density theorem of Khinchin and diophantine approximation","authors":"József Beck, William W. L. Chen","doi":"10.5802/jtnb.1255","DOIUrl":"https://doi.org/10.5802/jtnb.1255","url":null,"abstract":"The continuous version of a famous result of Khinchin says that a half-infinite torus line in the unit square [0,1] 2 exhibits superdensity, a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. We extend this result of Khinchin to the case when the unit torus [0,1] 2 is replaced by a finite polysquare translation surface, or square tiled surface. In particular, we show that it is possible to study this very number-theoretic problem by restricting to traditional tools in number theory, using only continued fractions and the famous 3-distance theorem in diophantine approximation combined with an iterative process.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider smooth projective curves C/𝔽 over a finite field and their symmetric squares C (2) . For a global function field K/𝔽, we study the K-rational points of C (2) . We describe the adelic points of C (2) surviving Frobenius descent and how the K-rational points fit there. Our methods also lead to an explicit bound on the number of K-rational points of C (2) satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.
{"title":"Rational points on symmetric squares of constant algebraic curves over function fields","authors":"Jennifer Berg, José Felipe Voloch","doi":"10.5802/jtnb.1252","DOIUrl":"https://doi.org/10.5802/jtnb.1252","url":null,"abstract":"We consider smooth projective curves C/𝔽 over a finite field and their symmetric squares C (2) . For a global function field K/𝔽, we study the K-rational points of C (2) . We describe the adelic points of C (2) surviving Frobenius descent and how the K-rational points fit there. Our methods also lead to an explicit bound on the number of K-rational points of C (2) satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The standard probability law on the set S(x,y) of y-friable integers not exceeding x assigns to each friable integer n a weight proportional to 1/n α , where α=α(x,y) is the saddle-point of the inverse Laplace integral for Ψ(x,y):=|S(x,y)|. This law presents a structural bias inasmuch it weights integers >x. We propose a quantitative measure of this bias and exhibit a related Gaussian distribution.
{"title":"Sur le biais d’une loi de probabilité relative aux entiers friables","authors":"Gérald Tenenbaum","doi":"10.5802/jtnb.1253","DOIUrl":"https://doi.org/10.5802/jtnb.1253","url":null,"abstract":"The standard probability law on the set S(x,y) of y-friable integers not exceeding x assigns to each friable integer n a weight proportional to 1/n α , where α=α(x,y) is the saddle-point of the inverse Laplace integral for Ψ(x,y):=|S(x,y)|. This law presents a structural bias inasmuch it weights integers >x. We propose a quantitative measure of this bias and exhibit a related Gaussian distribution.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a generalization of Pell’s equation, whose coefficients are certain algebraic integers. Let X 0 =0 and X n =2+X n-1 for each n∈ℤ ≥1 . We study the ℤ[X n-1 ]-solutions of the equation x 2 -X n 2 y 2 =1. By imitating the solution to the classical Pell’s equation, we introduce new continued fraction expansions for X n over ℤ[X n-1 ] and obtain an explicit solution of the generalized Pell’s equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber’s class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the ℤ 2 -extension over the rationals and show the convergence of the class numbers in ℤ 2 .
研究了系数为若干代数整数的Pell方程的推广。令每个n∈n≥1,X 0 =0, X n =2+X n-1。研究了方程x2 - xn2y2 =1的n [X n-1]-解。通过模拟经典Pell’s方程的解，引入了X n / n [X n-1]的新的连分式展开式，得到了广义Pell’s方程的显式解。此外，我们证明了当且仅当韦伯的类数问题的答案是肯定的，我们的显式解产生所有的解。我们还得到了素数在素数上扩展的类数之比的一个同余关系，并证明了素数在素数上的收敛性。
{"title":"Generalized Pell’s equations and Weber’s class number problem","authors":"Hyuga Yoshizaki","doi":"10.5802/jtnb.1249","DOIUrl":"https://doi.org/10.5802/jtnb.1249","url":null,"abstract":"We study a generalization of Pell’s equation, whose coefficients are certain algebraic integers. Let X 0 =0 and X n =2+X n-1 for each n∈ℤ ≥1 . We study the ℤ[X n-1 ]-solutions of the equation x 2 -X n 2 y 2 =1. By imitating the solution to the classical Pell’s equation, we introduce new continued fraction expansions for X n over ℤ[X n-1 ] and obtain an explicit solution of the generalized Pell’s equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber’s class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the ℤ 2 -extension over the rationals and show the convergence of the class numbers in ℤ 2 .","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136296166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new type of p-adic hypergeometric functions, which we call the p-adic hypergeometric functions of logarithmic type. The first main result is to prove the congruence relations that are similar to Dwork’s. The second main result is that the special values of our new functions appear in the syntomic regulators for hypergeometric curves, Fermat curves and some elliptic curves. According to the p-adic Beilinson conjecture by Perrin-Riou, they are expected to be related with the special values of p-adic L-functions. We provide one example for this.
{"title":"New p-adic hypergeometric functions and syntomic regulators","authors":"Masanori Asakura","doi":"10.5802/jtnb.1250","DOIUrl":"https://doi.org/10.5802/jtnb.1250","url":null,"abstract":"We introduce a new type of p-adic hypergeometric functions, which we call the p-adic hypergeometric functions of logarithmic type. The first main result is to prove the congruence relations that are similar to Dwork’s. The second main result is that the special values of our new functions appear in the syntomic regulators for hypergeometric curves, Fermat curves and some elliptic curves. According to the p-adic Beilinson conjecture by Perrin-Riou, they are expected to be related with the special values of p-adic L-functions. We provide one example for this.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article presents and specifies the main known results on the size of the subgroup of homotheties of ℓ-adic representations associated with the torsion of an abelian variety. Such estimates notably make it possible to give explicit uniform bounds in the framework of the Manin–Mumford problem.
{"title":"Homothéties explicites des représentations ℓ-adiques","authors":"Aurélien Galateau, César Martínez","doi":"10.5802/jtnb.1257","DOIUrl":"https://doi.org/10.5802/jtnb.1257","url":null,"abstract":"This article presents and specifies the main known results on the size of the subgroup of homotheties of ℓ-adic representations associated with the torsion of an abelian variety. Such estimates notably make it possible to give explicit uniform bounds in the framework of the Manin–Mumford problem.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136296035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Previous work of Kisin and Gee proves potential diagonalisability of two dimensional Barsotti–Tate representations of the Galois group of a finite extension K/ℚ p . In this paper we build upon their work by relaxing the Barsotti–Tate condition to one we call pseudo-Barsotti–Tate (which means that for certain embeddings κ:K→ℚ ¯ p we allow the κ-Hodge–Tate weights to be contained in [0,p] rather than [0,1]).
{"title":"Potential diagonalisability of pseudo-Barsotti–Tate representations","authors":"Robin Bartlett","doi":"10.5802/jtnb.1248","DOIUrl":"https://doi.org/10.5802/jtnb.1248","url":null,"abstract":"Previous work of Kisin and Gee proves potential diagonalisability of two dimensional Barsotti–Tate representations of the Galois group of a finite extension K/ℚ p . In this paper we build upon their work by relaxing the Barsotti–Tate condition to one we call pseudo-Barsotti–Tate (which means that for certain embeddings κ:K→ℚ ¯ p we allow the κ-Hodge–Tate weights to be contained in [0,p] rather than [0,1]).","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael A. Bennett, Philippe Michaud-Jacobs, Samir Siksek
for integers x,q,k,y and n, with k≥0 and n≥3. We extend work of the first and third-named authors by finding all solutions in the cases q=41 and q=97. We do this by constructing a Frey–Hellegouarch ℚ-curve defined over the real quadratic field K=ℚ(q), and using the modular method with multi-Frey techniques.
{"title":"ℚ-curves and the Lebesgue–Nagell equation","authors":"Michael A. Bennett, Philippe Michaud-Jacobs, Samir Siksek","doi":"10.5802/jtnb.1254","DOIUrl":"https://doi.org/10.5802/jtnb.1254","url":null,"abstract":"for integers x,q,k,y and n, with k≥0 and n≥3. We extend work of the first and third-named authors by finding all solutions in the cases q=41 and q=97. We do this by constructing a Frey–Hellegouarch ℚ-curve defined over the real quadratic field K=ℚ(q), and using the modular method with multi-Frey techniques.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We exhibit an explicit algorithm to compute three-point branched covers of the complex projective line when the uniformizing triangle group is Euclidean.
给出了一种计算均匀化三角形群为欧几里德时复射影线三点分支覆盖的显式算法。
{"title":"Computing Euclidean Belyi maps","authors":"Matthew Radosevich, John Voight","doi":"10.5802/jtnb.1256","DOIUrl":"https://doi.org/10.5802/jtnb.1256","url":null,"abstract":"We exhibit an explicit algorithm to compute three-point branched covers of the complex projective line when the uniformizing triangle group is Euclidean.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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