{"title":"Relative Non-cuspidality of Representations Induced from Split Parabolic Subgroups","authors":"S. Kato, K. Takano","doi":"10.3836/tjm/1502179309","DOIUrl":"https://doi.org/10.3836/tjm/1502179309","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46310051","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}
In this paper we enumerate nonhyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $ple 11$. Our algorithm works for nonhyperelliptic curves over an arbitrary finite field in characteristic $p ge 5$. We execute the algorithm for prime fields of $ple 11$ with our implementation on a computer algebra system Magma. Thanks to the fact that the cardinality of $mathbb{F}_{p^a}$-isomorphism classes of superspecial curves over $mathbb{F}_{p^a}$ of a fixed genus depends only on the parity of $a$, this paper contributes to the odd-degree case for genus $4$, whereas our previous paper contributes to the even-degree case.
{"title":"Computational Approach to Enumerate Non-hyperelliptic Superspecial Curves of Genus 4","authors":"Momonari Kudo, Shushi Harashita","doi":"10.3836/tjm/1502179310","DOIUrl":"https://doi.org/10.3836/tjm/1502179310","url":null,"abstract":"In this paper we enumerate nonhyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $ple 11$. Our algorithm works for nonhyperelliptic curves over an arbitrary finite field in characteristic $p ge 5$. We execute the algorithm for prime fields of $ple 11$ with our implementation on a computer algebra system Magma. Thanks to the fact that the cardinality of $mathbb{F}_{p^a}$-isomorphism classes of superspecial curves over $mathbb{F}_{p^a}$ of a fixed genus depends only on the parity of $a$, this paper contributes to the odd-degree case for genus $4$, whereas our previous paper contributes to the even-degree case.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89579666","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 present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton ⟨On groups of linear recurrences, I⟩ and by Aoki-Sakai ⟨Mod p equivalence classes of linear recurrence sequences of degree two⟩. Introduction The Lucas sequences, including the Fibonacci sequence, have been studied widely for a long time, and there is left an enormous accumulation of research. Particularly the divisibility problem is a main subject in the study on Lucas sequences. More explicitly, let P and Q be non-zero integers, and let (wk)k≥0 be the sequence defined by the linear recurrence relation wk+2 = Pwk+1 −Qwk with the intial terms w0, w1 ∈ Z. If w0 = 0 and w1 = 1, then (wk)k≥0 is nothing but the Lucas sequnces (Lk)k≥0 associated to (P,Q). The divisibility problem asks to describe the set {k ∈ N ; wk ≡ 0 mod m} for a positive integer m. The first step was certainly taken forward by Edouard Lucas [6] as the laws of apparition and repetition in the case where m is a prime number and (wk)k≥0 is the Lucas sequence, and there have been piled up various kinds of results after then. In this article we study the divisibility problem for Lucas sequences from a geometirc viewpoint, translating several descriptions on Lucas sequences into the language of affine group schemes. For example, the laws of apparition and repetition is formulated in our context as follows: Theorem(=Proposition 3.23+Theorem 3.25) Let P and Q be non-zero integers with (P,Q) = 1, and let w0, w1 ∈ Z with (w0, w1) = 1. Define the sequence (wk)k≥0 by the recurrence relation wk+2 = Pwk+1−Qwk with initial terms w0 and w1, and put μ = ordp(w 1−Pw0w1+Qw 0). Let p be an odd prime with (p,Q) = 1 and n a positive integer. Then we have the length of the orbit (w0 : w1)Θ in P(Z/pZ) = 1 (n ≤ μ) r(pn−μ) (n > μ) . Furthermore, there exists k ≥ 0 such that wk ≡ 0 mod pn if and only if (w0 : w1) ∈ (0 : 1).Θ in P1(Z/pnZ). Here Θ denotes the subgroup of G(D)(Z(p)) generated by β(θ) = (P/4Q, 1/4Q), and r(pν) denotes the rank mod pν of the Lucas sequence associated to (P,Q). ∗) Partially supported by Grant-in-Aid for Scientific Research No.26400024 2010 Mathematics Subject Classification Primary 13B05; Secondary 14L15, 12G05. 1
我们提出了一种在群方案理论的框架下看待Lucas序列的方法。这使我们能够从几何和函数的角度来处理Lucas序列，这是Laxton⟨关于线性递归组I⟩和Aoki-Sakai 10216提出的二阶线性递归序列的Mod p等价类。引言Lucas序列，包括Fibonacci序列，已经被广泛研究了很长一段时间，并且留下了巨大的研究积累。尤其是可分性问题是Lucas序列研究的一个主要课题。更明确地说，设P和Q是非零整数，设（wk）k≥0是线性递推关系wk+2=Pwk+1-Qwk与初始项w0，w1∈Z定义的序列。如果w0=0和w1=1，则（wk。整除性问题要求描述一个正整数m的集合{k∈N；wk lect 0 mod m}。Edouard Lucas[6]肯定地提出了第一步，即在m是素数且（wk）k≥0是Lucas序列的情况下的幻影和重复定律，此后出现了各种各样的结果。本文从几何的角度研究了Lucas序列的可分性问题，将Lucas序列上的一些描述转化为仿射群方案的语言。例如，幻影和重复定律在我们的上下文中被公式化如下：定理（=命题3.23+定理3.25）设P和Q是非零整数，其中（P，Q）=1，设w0，w1∈Z，其中（w0，w1）=1。通过具有初始项w0和w1的递推关系wk+2=Pwk+1−Qwk定义序列（wk）k≥0，并放入μ=ordp（w1−Pwww1+Qw0）。设p为奇素数，其中（p，Q）=1，n为正整数。然后我们得到了轨道的长度（w0:w1）Θ在P（Z/pZ）=1（n≤μ）r（pn-μ）（n＞μ）。此外，在P1（Z/pnZ）中存在k≥0使得wk≠0 mod pn当且仅当（w0:w1）∈（0:1）。这里，θ表示由β（θ）=（p/4Q，1/4Q）生成的G（D）（Z（p））的子群，r（pΓ）表示与（p，Q）相关的Lucas序列的秩mod pΓ。*）部分科研资助项目：No.2640024 2010数学学科分类小学13B05；次级14L15、12G05。1.
{"title":"Geometric Aspects of Lucas Sequences, I","authors":"Noriyuki Suwa","doi":"10.3836/TJM/1502179294","DOIUrl":"https://doi.org/10.3836/TJM/1502179294","url":null,"abstract":"We present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton ⟨On groups of linear recurrences, I⟩ and by Aoki-Sakai ⟨Mod p equivalence classes of linear recurrence sequences of degree two⟩. Introduction The Lucas sequences, including the Fibonacci sequence, have been studied widely for a long time, and there is left an enormous accumulation of research. Particularly the divisibility problem is a main subject in the study on Lucas sequences. More explicitly, let P and Q be non-zero integers, and let (wk)k≥0 be the sequence defined by the linear recurrence relation wk+2 = Pwk+1 −Qwk with the intial terms w0, w1 ∈ Z. If w0 = 0 and w1 = 1, then (wk)k≥0 is nothing but the Lucas sequnces (Lk)k≥0 associated to (P,Q). The divisibility problem asks to describe the set {k ∈ N ; wk ≡ 0 mod m} for a positive integer m. The first step was certainly taken forward by Edouard Lucas [6] as the laws of apparition and repetition in the case where m is a prime number and (wk)k≥0 is the Lucas sequence, and there have been piled up various kinds of results after then. In this article we study the divisibility problem for Lucas sequences from a geometirc viewpoint, translating several descriptions on Lucas sequences into the language of affine group schemes. For example, the laws of apparition and repetition is formulated in our context as follows: Theorem(=Proposition 3.23+Theorem 3.25) Let P and Q be non-zero integers with (P,Q) = 1, and let w0, w1 ∈ Z with (w0, w1) = 1. Define the sequence (wk)k≥0 by the recurrence relation wk+2 = Pwk+1−Qwk with initial terms w0 and w1, and put μ = ordp(w 1−Pw0w1+Qw 0). Let p be an odd prime with (p,Q) = 1 and n a positive integer. Then we have the length of the orbit (w0 : w1)Θ in P(Z/pZ) = 1 (n ≤ μ) r(pn−μ) (n > μ) . Furthermore, there exists k ≥ 0 such that wk ≡ 0 mod pn if and only if (w0 : w1) ∈ (0 : 1).Θ in P1(Z/pnZ). Here Θ denotes the subgroup of G(D)(Z(p)) generated by β(θ) = (P/4Q, 1/4Q), and r(pν) denotes the rank mod pν of the Lucas sequence associated to (P,Q). ∗) Partially supported by Grant-in-Aid for Scientific Research No.26400024 2010 Mathematics Subject Classification Primary 13B05; Secondary 14L15, 12G05. 1","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49485628","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 show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ frac{log{n}}{n}log{left|x-frac{P_n}{Q_n}right|}rightarrow -frac{pi^2}{3} quad text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.
{"title":"Diophantine Approximation by Negative Continued Fraction","authors":"Hiroaki Ito","doi":"10.3836/tjm/1502179364","DOIUrl":"https://doi.org/10.3836/tjm/1502179364","url":null,"abstract":"We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ frac{log{n}}{n}log{left|x-frac{P_n}{Q_n}right|}rightarrow -frac{pi^2}{3} quad text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44477331","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}
Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.
{"title":"On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of\u0000 Supersingular Abelian Varieties in $p$-adic Lie Extensions","authors":"M. Lim","doi":"10.3836/tjm/1502179341","DOIUrl":"https://doi.org/10.3836/tjm/1502179341","url":null,"abstract":"Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43452076","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 show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.
{"title":"On Weakly Reflective Submanifolds in Compact Isotropy Irreducible Riemannian Homogeneous Spaces","authors":"M. Morimoto","doi":"10.3836/tjm/1502179344","DOIUrl":"https://doi.org/10.3836/tjm/1502179344","url":null,"abstract":"We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47097020","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}
In this paper, we construct a right-angled 5-polytope P of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from P are locally rigid.
{"title":"Locally Rigid Right-angled Coxeter Groups with Fuchsian Ends in Dimension 5","authors":"Tomoshige Yukita","doi":"10.3836/tjm/1502179360","DOIUrl":"https://doi.org/10.3836/tjm/1502179360","url":null,"abstract":"In this paper, we construct a right-angled 5-polytope P of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from P are locally rigid.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45929492","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}
For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as linear mod $1$ transformations and generalized $beta$-transformations), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $beta$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith problem on the density of periodic measures.
{"title":"Irregular Sets for Piecewise Monotonic Maps","authors":"Yushi Nakano, Kenichiro Yamamoto","doi":"10.3836/tjm/1502179349","DOIUrl":"https://doi.org/10.3836/tjm/1502179349","url":null,"abstract":"For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as linear mod $1$ transformations and generalized $beta$-transformations), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $beta$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith problem on the density of periodic measures.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41347528","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}
{"title":"Quiver Representations, Group Characters, and Prime Graphs of Finite Groups","authors":"N. Iiyori, M. Sawabe","doi":"10.3836/TJM/1502179297","DOIUrl":"https://doi.org/10.3836/TJM/1502179297","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41681375","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}
In this paper, we introduce the higher order hypergeometric Euler numbers and show several interesting expressions. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. One advantage of hypergeometric numbers, including Bernoulli, Cauchy and Euler hypergeometric numbers, is the natural extension of determinant expressions of the numbers. As applications, we can get the inversion relations such that Euler numbers are elements in the determinant.
{"title":"Several Properties of Multiple Hypergeometric Euler Numbers","authors":"T. Komatsu, Wenpeng Zhang","doi":"10.3836/TJM/1502179290","DOIUrl":"https://doi.org/10.3836/TJM/1502179290","url":null,"abstract":"In this paper, we introduce the higher order hypergeometric Euler numbers and show several interesting expressions. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. One advantage of hypergeometric numbers, including Bernoulli, Cauchy and Euler hypergeometric numbers, is the natural extension of determinant expressions of the numbers. As applications, we can get the inversion relations such that Euler numbers are elements in the determinant.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42382968","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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