Let q be a prime with q ≡ 7 mod 8, and let K = Q( √ −q). Then 2 splits in K, and we write p for either of the primes K above 2. Let K∞ be the unique Z2-extension of K unramified outside p with n-th layer Kn. For certain quadratic and biquadratic extensions F/K, we prove a simple exact formula for the λ-invariant of the Galois group of the maximal abelian 2-extension unramified outside p of the field F∞ = FK∞. Equivalently, our result determines the exact Z2-corank of certain Selmer groups over F∞ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to K of the Gross curve with complex multiplication defined over the Hilbert class field of K. We also discuss computations of the associated Selmer groups over Kn in the case when the λ-invariant is equal to 1.
设q为素数，q≠7 mod 8，设K=q（√−q）。然后2在K中分裂，我们为2以上的素数K中的任何一个写p。设K∞是在p外具有第n层Kn的K的唯一Z2扩张。对于某些二次和双二次扩张F/K，我们证明了域F∞=FK∞的p外最大阿贝尔2-扩张的Galois群的λ-不变量的一个简单精确公式。等价地，我们的结果确定了具有复乘法的高维阿贝尔变种的一大族二次扭曲的F∞上某些Selmer群的精确Z2 corank，这是在K的Hilbert类域上定义的具有复乘法Gross曲线的标量对K的限制。我们还讨论了当λ-不变量等于1时，Kn上的相关Selmer群的计算。
{"title":"On the $lambda$-invariant of Selmer Groups Arising from Certain Quadratic Twists of Gross Curves","authors":"Jianing Li","doi":"10.3836/tjm/1502179379","DOIUrl":"https://doi.org/10.3836/tjm/1502179379","url":null,"abstract":"Let q be a prime with q ≡ 7 mod 8, and let K = Q( √ −q). Then 2 splits in K, and we write p for either of the primes K above 2. Let K∞ be the unique Z2-extension of K unramified outside p with n-th layer Kn. For certain quadratic and biquadratic extensions F/K, we prove a simple exact formula for the λ-invariant of the Galois group of the maximal abelian 2-extension unramified outside p of the field F∞ = FK∞. Equivalently, our result determines the exact Z2-corank of certain Selmer groups over F∞ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to K of the Gross curve with complex multiplication defined over the Hilbert class field of K. We also discuss computations of the associated Selmer groups over Kn in the case when the λ-invariant is equal to 1.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44368783","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}
Minoru Tanaka, T. Akamatsu, R. Sinclair, M. Yamaguchi
There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids and tori are now known. Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family {Mn}n of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or maximum of the Gaussian curvature function on the meridian goes to infinity as n tends to infinity. Thus, our family includes surfaces which have arbitrarily many bands of alternately increasing or decreasing Gaussian curvature, although each member of this family has a simple cut locus structure.
{"title":"A New Family of Latitudinally Corrugated Two-spheres of Revolution with Simple Cut Locus Structure","authors":"Minoru Tanaka, T. Akamatsu, R. Sinclair, M. Yamaguchi","doi":"10.3836/tjm/1502179366","DOIUrl":"https://doi.org/10.3836/tjm/1502179366","url":null,"abstract":"There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids and tori are now known. Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family {Mn}n of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or maximum of the Gaussian curvature function on the meridian goes to infinity as n tends to infinity. Thus, our family includes surfaces which have arbitrarily many bands of alternately increasing or decreasing Gaussian curvature, although each member of this family has a simple cut locus structure.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41307926","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 define a syntomic complex for modulus pair $(X,D)$, where $X$ is a regular semi-stable family and $D$ is an effective Cartier divisor on $X$ and we compute its cohomology sheaves. We construct a symbol map for this syntomic complex for modulus pair $(X,D)$ and investigate its cokernel by computing its cohomology sheaves. To compute the structure of the cohomology sheaves of syntomic complex with modulus, we define some filtrations on it.
{"title":"On Syntomic Complexes with Modulus for Semi-stable Reduction Cases","authors":"Kento Yamamoto","doi":"10.3836/tjm/1502179342","DOIUrl":"https://doi.org/10.3836/tjm/1502179342","url":null,"abstract":"In this paper, we define a syntomic complex for modulus pair $(X,D)$, where $X$ is a regular semi-stable family and $D$ is an effective Cartier divisor on $X$ and we compute its cohomology sheaves. We construct a symbol map for this syntomic complex for modulus pair $(X,D)$ and investigate its cokernel by computing its cohomology sheaves. To compute the structure of the cohomology sheaves of syntomic complex with modulus, we define some filtrations on it.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43028180","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 $G$ be the group $SL(2,mathbb{R})$, $Psubset G$ be the parabolic subgroup of upper triangular matrices and $Gammasubset G$ be a cocompact lattice. A right action of $P$ on $Gammabackslash G$ defines an orbit foliation $mathcal{F}_P$. We compute the leafwise cohomology ring $H^*(mathcal{F}_P)$ by exploiting non-abelian harmonic analysis on $G$.
{"title":"Computation of Some Leafwise Cohomology Ring","authors":"S. Mori","doi":"10.3836/tjm/1502179396","DOIUrl":"https://doi.org/10.3836/tjm/1502179396","url":null,"abstract":"Let $G$ be the group $SL(2,mathbb{R})$, $Psubset G$ be the parabolic subgroup of upper triangular matrices and $Gammasubset G$ be a cocompact lattice. A right action of $P$ on $Gammabackslash G$ defines an orbit foliation $mathcal{F}_P$. We compute the leafwise cohomology ring $H^*(mathcal{F}_P)$ by exploiting non-abelian harmonic analysis on $G$.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48424965","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":"A Multi-parameter Family of Self-avoiding Walks on the\u0000 Sierpiński Gasket","authors":"T. Otsuka","doi":"10.3836/TJM/1502179338","DOIUrl":"https://doi.org/10.3836/TJM/1502179338","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48323683","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 investigate properties of orbits of Hermann actions as submanifolds without assuming the commutability of involutions which define Hermann actions. In particular, we compute the second fundamental form of orbits of Hermann action, and give a sufficient condition for orbits of Hermann action to be weakly reflective (resp. arid) submanifolds.
{"title":"Geometric Properties of Orbits of Hermann actions","authors":"S. Ohno","doi":"10.3836/tjm/1502179367","DOIUrl":"https://doi.org/10.3836/tjm/1502179367","url":null,"abstract":"In this paper, we investigate properties of orbits of Hermann actions as submanifolds without assuming the commutability of involutions which define Hermann actions. In particular, we compute the second fundamental form of orbits of Hermann action, and give a sufficient condition for orbits of Hermann action to be weakly reflective (resp. arid) submanifolds.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43824839","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":"Construction of Solutions of the Classical Field Equation of a Massless Klein-Gordon Field Interacting with a Static Source","authors":"Toshimitsu Takaesu","doi":"10.3836/TJM/1502179328","DOIUrl":"https://doi.org/10.3836/TJM/1502179328","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70165763","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 $p=2^{e+1}q+1$ be an odd prime number with $2 nmid q$. Let $K$ be the imaginary cyclic field of conductor $p$ and degree $2^{e+1}$. We denote by $mathcal{F}$ the imaginary quadratic subextension of the imaginary $(2,,2)$-extension $K(sqrt{2})/K^+$ with $mathcal{F} neq K$. We determine the Galois module structure of the $2$-part of the class group of $mathcal{F}$.
{"title":"On the Class Group of an Imaginary Cyclic Field of Conductor $8p$ and $2$-power Degree","authors":"H. Ichimura, Hiroki Sumida-Takahashi","doi":"10.3836/TJM/1502179326","DOIUrl":"https://doi.org/10.3836/TJM/1502179326","url":null,"abstract":"Let $p=2^{e+1}q+1$ be an odd prime number with $2 nmid q$. Let $K$ be the imaginary cyclic field of conductor $p$ and degree $2^{e+1}$. We denote by $mathcal{F}$ the imaginary quadratic subextension of the imaginary $(2,,2)$-extension $K(sqrt{2})/K^+$ with $mathcal{F} neq K$. We determine the Galois module structure of the $2$-part of the class group of $mathcal{F}$.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70165720","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":"Contracted Ideals of $p$-adic Integral Group Rings","authors":"Joongul Lee","doi":"10.3836/tjm/1502179313","DOIUrl":"https://doi.org/10.3836/tjm/1502179313","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48266253","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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