Journal of the Mathematical Society of Japan Articles (Project Euclid)
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Toy models for D. H. Lehmer's conjecture
http://projecteuclid.org/euclid.jmsj/1280496816
<strong>Eiichi BANNAI</strong>, <strong>Tsuyoshi MIEZAKI</strong><p><strong>Source: </strong>J. Math. Soc. Japan, Volume 62, Number 3, 687--705.</p><p><strong>Abstract:</strong><br/>
In 1947, Lehmer conjectured that the Ramanujan τ-function τ( m ) never vanishes for all positive integers m , where τ( m ) are the Fourier coefficients of the cusp form Δ 24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 10 15 , and in 1999, Jordan and Kelly for m < 22689242781695999.
The theory of spherical t -design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's τ-function gives the coefficients of a weighted theta series of the E 8 -lattice. It is shown, by Venkov, de la Harpe, and Pache, that τ( m ) = 0 is equivalent to the fact that the shell of norm 2 m of the E 8 -lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical t -design.
Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the m -th Fourier coefficient of the weighted theta series of the Z 2 -lattice and the A 2 -lattice does not vanish, when the shell of norm m of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the Z 2 -lattice (resp. A 2 -lattice).
</p>projecteuclid.org/euclid.jmsj/1280496816_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTRelative stability associated to quantised extremal Kähler metricshttps://projecteuclid.org/euclid.jmsj/1556179398<strong>Yoshinori HASHIMOTO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 861--880.</p><p><strong>Abstract:</strong><br/>
We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and relative $K$-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa–Székelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann–Roch theorem.
</p>projecteuclid.org/euclid.jmsj/1556179398_20190724220337Wed, 24 Jul 2019 22:03 EDT$L_p$ regularity theorem for elliptic equations in less smooth domainshttps://projecteuclid.org/euclid.jmsj/1556092821<strong>Yoichi MIYAZAKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 881--907.</p><p><strong>Abstract:</strong><br/>
We consider a $2m$th-order strongly elliptic operator $A$ subject to Dirichlet boundary conditions in a domain $\Omega$ of $\mathbb{R}^{n}$, and show the $L_{p}$ regularity theorem, assuming that the domain has less smooth boundary. We derive the regularity theorem from the following isomorphism theorems in Sobolev spaces. Let $k$ be a nonnegative integer. When $A$ is a divergence form elliptic operator, $A-\lambda$ has a bounded inverse from the Sobolev space $W^{k-m}_{p}(\Omega)$ into $W^{k+m}_{p}(\Omega)$ for $\lambda$ belonging to a suitable sectorial region of the complex plane, if $\Omega$ is a uniformly $C^{k,1}$ domain. When $A$ is a non-divergence form elliptic operator, $A-\lambda$ has a bounded inverse from $W^{k}_{p}(\Omega)$ into $W^{k+2m}_{p}(\Omega)$, if $\Omega$ is a uniformly $C^{k+m,1}$ domain. Compared with the known results, we weaken the smoothness assumption on the boundary of $\Omega$ by $m-1$.
</p>projecteuclid.org/euclid.jmsj/1556092821_20190724220337Wed, 24 Jul 2019 22:03 EDTOn sharper estimates of Ohsawa–Takegoshi $L^2$-extension theoremhttps://projecteuclid.org/euclid.jmsj/1552377784<strong>Genki HOSONO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 909--914.</p><p><strong>Abstract:</strong><br/>
We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. When the Hartogs domain defined by the weight function is strictly pseudoconvex, this estimate is strictly sharper than known optimal estimates. When the weight function is radial, we prove that our estimate provides the $L^2$-minimum extension.
</p>projecteuclid.org/euclid.jmsj/1552377784_20190724220337Wed, 24 Jul 2019 22:03 EDTRough flowshttps://projecteuclid.org/euclid.jmsj/1559030414<strong>Ismaël BAILLEUL</strong>, <strong>Sebastian RIEDEL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 915--978.</p><p><strong>Abstract:</strong><br/>
We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semimartingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for ‘Gaussian’ stochastic flows.
</p>projecteuclid.org/euclid.jmsj/1559030414_20190724220337Wed, 24 Jul 2019 22:03 EDTGeneral formal solutions for a unified family of $P_{\mathrm{J}}$-hierarchies (J=I, II, IV, 34)https://projecteuclid.org/euclid.jmsj/1556092820<strong>Yoko UMETA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 979--1003.</p><p><strong>Abstract:</strong><br/>
A unified family of $P_{\mathrm{J}}$-hierarchies (J=I, II, IV, 34) with a large parameter is introduced and we construct general formal solutions which are called instanton-type solutions for the system.
</p>projecteuclid.org/euclid.jmsj/1556092820_20190724220337Wed, 24 Jul 2019 22:03 EDTOn an upper bound of $\lambda$-invariants of $\mathbb{Z}_p$-extensions over an imaginary quadratic fieldhttps://projecteuclid.org/euclid.jmsj/1556179397<strong>Kazuaki MURAKAMI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 1005--1026.</p><p><strong>Abstract:</strong><br/>
For an odd prime number $p$, we give an explicit upper bound of $\lambda$-invariants for all $\mathbb{Z}_p$-extensions of an imaginary quadratic field $k$ under several assumptions. We also give an explicit upper bound of $\lambda$-invariants for all $\mathbb{Z}_p$-extensions of $k$ in the case where the $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $k$ is equal to 3.
</p>projecteuclid.org/euclid.jmsj/1556179397_20190724220337Wed, 24 Jul 2019 22:03 EDTSolomon–Terao algebra of hyperplane arrangementshttps://projecteuclid.org/euclid.jmsj/1563350426<strong>Takuro ABE</strong>, <strong>Toshiaki MAENO</strong>, <strong>Satoshi MURAI</strong>, <strong>Yasuhide NUMATA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1027--1047.</p><p><strong>Abstract:</strong><br/>
We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon–Terao algebra $ST(\mathcal{A}, \eta)$, where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $ST(\mathcal{A}, \eta)$ is Artinian when $\eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon–Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $ST(\mathcal{A}, \eta)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials of $ST(\mathcal{A}, \eta)$ when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.
</p>projecteuclid.org/euclid.jmsj/1563350426_20191024220345Thu, 24 Oct 2019 22:03 EDTOptimal problem for mixed $p$-capacitieshttps://projecteuclid.org/euclid.jmsj/1560391345<strong>Baocheng ZHU</strong>, <strong>Xiaokang LUO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1049--1079.</p><p><strong>Abstract:</strong><br/>
In this paper, the optimal problem for mixed $p$-capacities is investigated. The Orlicz and $L_q$ geominimal $p$-capacities are proposed and their properties, such as invariance under orthogonal matrices, isoperimetric type inequalities and cyclic type inequalities are provided as well. Moreover, the existence of the $p$-capacitary Orlicz–Petty bodies for multiple convex bodies is established, and the Orlicz and $L_q$ mixed geominimal $p$-capacities for multiple convex bodies are introduced. The continuity of the Orlicz mixed geominimal $p$-capacities and some isoperimetric type inequalities of the $L_q$ mixed geominimal $p$-capacities are proved.
</p>projecteuclid.org/euclid.jmsj/1560391345_20191024220345Thu, 24 Oct 2019 22:03 EDTObtuse constants of Alexandrov spaceshttps://projecteuclid.org/euclid.jmsj/1560391344<strong>Ayato MITSUISHI</strong>, <strong>Takao YAMAGUCHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1081--1103.</p><p><strong>Abstract:</strong><br/>
We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to $\pi/2$, where we prove some rigidity for spaces. Although we consider Alexandrov spaces with curvature bounded below, the results are new even in the Riemannian case.
</p>projecteuclid.org/euclid.jmsj/1560391344_20191024220345Thu, 24 Oct 2019 22:03 EDTHeight, trunk and representativity of knotshttps://projecteuclid.org/euclid.jmsj/1562724416<strong>Ryan BLAIR</strong>, <strong>Makoto OZAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1105--1121.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a counterexample for the conjecture which states that the height is additive under connected sum of knots. We also define the minimal height of a knot and give a potential example which has a gap between the height and the minimal height. Next, we show that the representativity is bounded above by a half of the trunk. We also define the trunk of a tangle and show that if a knot has an essential tangle decomposition, then the representativity is bounded above by half of the trunk of either of the two tangles. Finally, we remark on the difference among Gabai's thin position, ordered thin position and minimal critical position. We also give an example of a knot which bounds an essential non-orientable spanning surface, but has arbitrarily large representativity.
</p>projecteuclid.org/euclid.jmsj/1562724416_20191024220345Thu, 24 Oct 2019 22:03 EDTDeeply concatenable subgroups might never be freehttps://projecteuclid.org/euclid.jmsj/1553068892<strong>Samuel M. CORSON</strong>, <strong>Saharon SHELAH</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1123--1136.</p><p><strong>Abstract:</strong><br/>
We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer–Specker group $\mathbb{Z}^{\omega}$ and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented.
</p>projecteuclid.org/euclid.jmsj/1553068892_20191024220345Thu, 24 Oct 2019 22:03 EDTThe hyperbolic-type point processhttps://projecteuclid.org/euclid.jmsj/1553068884<strong>Nizar DEMNI</strong>, <strong>Pierre LAZAG</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1137--1152.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a two-parameters determinantal point process in the Poincaré disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r \rightarrow 1^-$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc.
</p>projecteuclid.org/euclid.jmsj/1553068884_20191024220345Thu, 24 Oct 2019 22:03 EDTProof of Kobayashi's rank conjecture on Clifford–Klein formshttps://projecteuclid.org/euclid.jmsj/1562140845<strong>Yosuke MORITA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1153--1171.</p><p><strong>Abstract:</strong><br/>
Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space $G/H$ of reductive type does not admit a compact Clifford–Klein form if $\operatorname{rank} G - \operatorname{rank} K < \operatorname{rank} H - \operatorname{rank} K_H$. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford–Klein forms proved previously by the author, and use the Sullivan model for a reductive pair due to Cartan–Chevalley–Koszul–Weil.
</p>projecteuclid.org/euclid.jmsj/1562140845_20191024220345Thu, 24 Oct 2019 22:03 EDTA proof of Saitoh's conjecture for conjugate Hardy $H^{2}$ kernelshttps://projecteuclid.org/euclid.jmsj/1563350422<strong>Qi'an GUAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1173--1179.</p><p><strong>Abstract:</strong><br/>
In this article, we obtain a strict inequality between the conjugate Hardy $H^{2}$ kernels and the Bergman kernels on planar regular regions with $n > 1$ boundary components, which is a conjecture of Saitoh.
</p>projecteuclid.org/euclid.jmsj/1563350422_20191024220345Thu, 24 Oct 2019 22:03 EDTNumerically trivial automorphisms of Enriques surfaces in characteristic 2https://projecteuclid.org/euclid.jmsj/1553068887<strong>Igor DOLGACHEV</strong>, <strong>Gebhard MARTIN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1181--1200.</p><p><strong>Abstract:</strong><br/>
An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second cohomology group (this group modulo torsion subgroup). Extending the results of Mukai and Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\operatorname{Aut}_{\operatorname{ct}}(S)$ of an Enriques surface $S$ is of order $\le 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\mathrm{Aut}_{\operatorname{ct}}(S)$ is a cyclic group of odd order $n \in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order 8 and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\mathrm{Aut}_{\operatorname{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\le 4$ unless $p = 2$, where $\mathrm{Aut}_{\operatorname{nt}}(S)$ is a subgroup of a 2-elementary group of rank $\le 2$.
</p>projecteuclid.org/euclid.jmsj/1553068887_20191024220345Thu, 24 Oct 2019 22:03 EDTBifurcation sets of real polynomial functions of two variables and Newton polygonshttps://projecteuclid.org/euclid.jmsj/1560412821<strong>Masaharu ISHIKAWA</strong>, <strong>Tat-Thang NGUYEN</strong>, <strong>Tien-Son PHẠM</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1201--1222.</p><p><strong>Abstract:</strong><br/>
In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called “cleaving” and “vanishing”, in the same setting. Finally, we give an upper bound of the number of atypical values at infinity in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively.
</p>projecteuclid.org/euclid.jmsj/1560412821_20191024220345Thu, 24 Oct 2019 22:03 EDTGeneralizations of the Conway–Gordon theorems and intrinsic knotting on complete graphshttps://projecteuclid.org/euclid.jmsj/1560499222<strong>Hiroko MORISHITA</strong>, <strong>Ryo NIKKUNI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1223--1241.</p><p><strong>Abstract:</strong><br/>
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway–Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway–Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph whose number of vertices is greater than or equal to six, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.
</p>projecteuclid.org/euclid.jmsj/1560499222_20191024220345Thu, 24 Oct 2019 22:03 EDTHardy and Rellich inequalities with exact missing terms on homogeneous groupshttps://projecteuclid.org/euclid.jmsj/1560412820<strong>Duy Tuan NGUYEN</strong>, <strong>Nguyen LAM-HOANG</strong>, <strong>Triet Anh NGUYEN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1243--1256.</p><p><strong>Abstract:</strong><br/>
We prove several identities on homogeneous groups that imply the Hardy and Rellich inequalities for Bessel pairs. These equalities give a straightforward understanding of some of the Hardy and Rellich inequalities as well as the absence of nontrivial optimizers and the existence/nonexistence of “virtual”extremizers.
</p>projecteuclid.org/euclid.jmsj/1560412820_20191024220345Thu, 24 Oct 2019 22:03 EDTArnold's problem on monotonicity of the Newton number for surface singularitieshttps://projecteuclid.org/euclid.jmsj/1560499224<strong>Szymon BRZOSTOWSKI</strong>, <strong>Tadeusz KRASIŃSKI</strong>, <strong>Justyna WALEWSKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1257--1268.</p><p><strong>Abstract:</strong><br/>
According to the Kouchnirenko Theorem, for a generic (meaning non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $\mu (f)$ is equal to the Newton number $\nu (\mathbf{\Gamma}_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $\mathbf{\Gamma}_+ (f)$. We give a simple condition characterizing, in terms of $\mathbf{\Gamma}_+ (f)$ and $\mathbf{\Gamma}_+ (g)$, the equality $\nu (\mathbf{\Gamma}_{+}(f)) = \nu (\mathbf{\Gamma}_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $\mathbf{\Gamma}_+ (f) \subset \mathbf{\Gamma}_+ (g)$. This is a complete solution to an Arnold problem (No. 1982-16 in his list of problems) in this case.
</p>projecteuclid.org/euclid.jmsj/1560499224_20191024220345Thu, 24 Oct 2019 22:03 EDTResidually faithful modules and the Cohen–Macaulay type of idealizationshttps://projecteuclid.org/euclid.jmsj/1560499223<strong>Shiro GOTO</strong>, <strong>Shinya KUMASHIRO</strong>, <strong>Nguyen Thi Hong LOAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1269--1291.</p><p><strong>Abstract:</strong><br/>
The Cohen–Macaulay type of idealizations of maximal Cohen–Macaulay modules over Cohen–Macaulay local rings is closely explored. There are two extremal cases, one of which is related to the theory of Ulrich modules, and the other one is related to the theory of residually faithful modules and closed ideals, developed by Brennan and Vasconcelos.
</p>projecteuclid.org/euclid.jmsj/1560499223_20191024220345Thu, 24 Oct 2019 22:03 EDTMaximal regularity of the Stokes system with Navier boundary condition in general unbounded domainshttps://projecteuclid.org/euclid.jmsj/1563847517<strong>Reinhard FARWIG</strong>, <strong>Veronika ROSTECK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1293--1319.</p><p><strong>Abstract:</strong><br/>
Consider the instationary Stokes system in general unbounded domains $\Omega \subset \mathbb{R}^n$, $n \geq 2$, with boundary of uniform class $C^3$, and Navier slip or Robin boundary condition. The main result of this article is the maximal regularity of the Stokes operator in function spaces of the type $\tilde{L}^q$ defined as $L^q \cap L^2$ when $q \geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain.
</p>projecteuclid.org/euclid.jmsj/1563847517_20191024220345Thu, 24 Oct 2019 22:03 EDTIsotropic quadrangular algebrashttps://projecteuclid.org/euclid.jmsj/1562033194<strong>Bernhard MÜHLHERR</strong>, <strong>Richard M. WEISS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 4, 1321--1380.</p><p><strong>Abstract:</strong><br/>
Quadrangular algebras arise in the theory of Tits quadrangles. They are anisotropic if and only if the corresponding Tits quadrangle is, in fact, a Moufang quadrangle. Anisotropic quadrangular algebras were classified in the course of classifying Moufang polygons. In this paper we extend the classification of anisotropic quadrangular algebras to a classification of isotropic quadrangular algebras satisfying a natural non-degeneracy condition.
</p>projecteuclid.org/euclid.jmsj/1562033194_20191024220345Thu, 24 Oct 2019 22:03 EDTCombinatorics of double loop suspensions, evaluation maps and Cohen groupshttps://projecteuclid.org/euclid.jmsj/1579078876<strong>Ruizhi HUANG</strong>, <strong>Jie WU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 43 pages.</p><p><strong>Abstract:</strong><br/>
We reformulate Milgram's model of a double loop suspension in terms of a preoperad of posets, each stage of which is the poset of all ordered partitions of a finite set. Using this model, we give a combinatorial model for the evaluation map and use it to study the Cohen representation for the group of homotopy classes of maps between double loop suspensions. Demonstrating the general theory, we recover Wu's shuffle relations and further provide a type of secondary relations in Cohen groups by using Toda brackets. In particular, we prove certain maps are null-homotopic by combining our relations and the classical James–Hopf invariants.
</p>projecteuclid.org/euclid.jmsj/1579078876_20200115040154Wed, 15 Jan 2020 04:01 ESTTwo-weighted estimates for positive operators and Doob maximal operators on filtered measure spaceshttps://projecteuclid.org/euclid.jmsj/1576486817<strong>Wei CHEN</strong>, <strong>Chunxiang ZHU</strong>, <strong>Yahui ZUO</strong>, <strong>Yong JIAO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and Uriarte-Tuero [ 30 ]. Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen–Pérez type and Lerner–Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.
</p>projecteuclid.org/euclid.jmsj/1576486817_20200115040154Wed, 15 Jan 2020 04:01 ESTDimensions of multi-fan duality algebrashttps://projecteuclid.org/euclid.jmsj/1575536414<strong>Anton AYZENBERG</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. The algebras constructed this way include many important examples: cohomology algebras of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced and studied by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured that the same holds for any simplicial cycle. We disprove this conjecture by showing that the colors of singular points of the cycle may affect the dimensions. However, the colors of nonsingular points are irrelevant. By using bistellar moves we show that the number of distinct dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial on general pseudomanifolds.
</p>projecteuclid.org/euclid.jmsj/1575536414_20200115040154Wed, 15 Jan 2020 04:01 ESTUnitary $t$-groupshttps://projecteuclid.org/euclid.jmsj/1574672464<strong>Eiichi BANNAI</strong>, <strong>Gabriel NAVARRO</strong>, <strong>Noelia RIZO</strong>, <strong>Pham Huu TIEP</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
Relying on the main results of [ GT ], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary 4-group, which is also a unitary 5-group, but not a unitary $t$-group for any $t \geq 6$.
</p>projecteuclid.org/euclid.jmsj/1574672464_20200115040154Wed, 15 Jan 2020 04:01 ESTWidths of highly excited resonances in multidimensional molecular predissociationhttps://projecteuclid.org/euclid.jmsj/1574672465<strong>André MARTINEZ</strong>, <strong>Vania SORDONI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 44 pages.</p><p><strong>Abstract:</strong><br/>
We investigate the simple resonances of a 2 by 2 matrix of $n$-dimensional semiclassical Schrödinger operators that interact through a first order differential operator. We assume that one of the two (analytic) potentials admits a well with non empty interior, while the other one is non trapping and creates a barrier between the well and infinity. Under a condition on the resonant state inside the well, we find an optimal lower bound on the width of the resonance. The method of proof relies on Carleman estimates, microlocal propagation of the microsupport, and a refined study of a non involutive double characteristic problem in the framework of Sjöstrand's analytic microlocal theory.
</p>projecteuclid.org/euclid.jmsj/1574672465_20200115040154Wed, 15 Jan 2020 04:01 ESTOn superspecial abelian surfaces over finite fields IIhttps://projecteuclid.org/euclid.jmsj/1574672466<strong>Jiangwei XUE</strong>, <strong>Tse-Chung YANG</strong>, <strong>Chia-Fu YU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 29 pages.</p><p><strong>Abstract:</strong><br/>
Extending the results of the current authors [Doc. Math., 21 (2016), 1607–1643] and [Asian J. Math. to appear, arXiv:1404.2978], we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field $\mathbb{F}_p$. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.
</p>projecteuclid.org/euclid.jmsj/1574672466_20200115040154Wed, 15 Jan 2020 04:01 ESTMilnor–Hamm sphere fibrations and the equivalence problemhttps://projecteuclid.org/euclid.jmsj/1574154014<strong>Raimundo N. ARAÚJO DOS SANTOS</strong>, <strong>Maico F. RIBEIRO</strong>, <strong>Mihai TIBĂR</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We introduce the sphere fibration for real map germs with radial discriminant and we address the problem of its equivalence with the Milnor–Hamm tube fibration. Under natural conditions, we prove the existence of open book structures with singularities and solve the equivalence problem.
</p>projecteuclid.org/euclid.jmsj/1574154014_20200115040154Wed, 15 Jan 2020 04:01 ESTIntuitive representation of local cohomology groupshttps://projecteuclid.org/euclid.jmsj/1573786819<strong>Daichi KOMORI</strong>, <strong>Kohei UMETA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 29 pages.</p><p><strong>Abstract:</strong><br/>
We construct a framework which gives intuitive representation of local cohomology groups. By defining the concrete mappings among them, we show their equivalence. As an application, we justify intuitive representation of Laplace hyperfunctions.
</p>projecteuclid.org/euclid.jmsj/1573786819_20200115040154Wed, 15 Jan 2020 04:01 ESTClassifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$https://projecteuclid.org/euclid.jmsj/1573786830<strong>Osamu IYAMA</strong>, <strong>Xiaojin ZHANG</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 34 page.</p><p><strong>Abstract:</strong><br/>
We build a bijection between the set ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda$ of isomorphism classes of basic support $\tau$-tilting modules over the Auslander algebra $\Lambda$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set ${\rm tilt}\hspace{.01in} \Lambda$ of isomorphism classes of basic tilting $\Lambda$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $\Gamma$ of Dynkin type $A_n$ as a factor algebra of $\Lambda$, we show that the tensor functor $-\otimes_{\Lambda} \Gamma$ induces a bijection between ${\rm s}\tau\mbox{-tilt}\hspace{.01in} \Lambda\to {\rm s}\tau\mbox{-tilt}\hspace{.01in} \Gamma$. This recover Mizuno's anti-isomorphism $\mathfrak{S}_{n+1} \to {\rm s}\tau\mbox{-tilt}\hspace{.01in} \Gamma$ of posets for type $A_n$.
</p>projecteuclid.org/euclid.jmsj/1573786830_20200115040154Wed, 15 Jan 2020 04:01 ESTChow rings of versal complete flag varietieshttps://projecteuclid.org/euclid.jmsj/1573636129<strong>Nobuaki YAGITA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 39 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we try to compute Chow rings of versal complete flag varieties corresponding to simple Lie groups, by using generalized Rost motives. As applications, we give new proofs of Totaro's results for the torsion indexes of simple Lie groups except for spin groups.
</p>projecteuclid.org/euclid.jmsj/1573636129_20200115040154Wed, 15 Jan 2020 04:01 ESTExplicit constructions of bordism of Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1 \times \mathbb{C} P^{n-1}$https://projecteuclid.org/euclid.jmsj/1573182015<strong>Grigory SOLOMADIN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In the present paper we construct two new explicit complex bordisms between any two projective bundles over $\mathbb{C} P^1$ of the same complex dimension, including the Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1 \times \mathbb{C} P^{n-1}$. These constructions reduce the bordism problem to the null-bordism of some projective bundle over $\mathbb{C} P^1$ with the non-standard stably complex structure.
</p>projecteuclid.org/euclid.jmsj/1573182015_20200115040154Wed, 15 Jan 2020 04:01 ESTJoint denseness of Hurwitz zeta functions with algebraic irrational parametershttps://projecteuclid.org/euclid.jmsj/1573182016<strong>Yoonbok LEE</strong>, <strong>Hidehiko MISHOU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the joint denseness of the Riemann zeta function and Hurwitz zeta functions with certain algebraic irrational and transcendental parameters on $\Re s > 1$. We also provide evidence for the denseness of the Hurwitz zeta function with an algebraic irrational parameter on $1/2 < \Re s < 1$.
</p>projecteuclid.org/euclid.jmsj/1573182016_20200115040154Wed, 15 Jan 2020 04:01 ESTDiagram automorphisms and quantum groupshttps://projecteuclid.org/euclid.jmsj/1572660116<strong>Toshiaki SHOJI</strong>, <strong>Zhiping ZHOU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 33 pages.</p><p><strong>Abstract:</strong><br/>
Let $\mathbf{U}^-_q = \mathbf{U}^-_q(\mathfrak{g})$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, and $\sigma : \mathfrak{g} \to \mathfrak{g}$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak{g}^{\sigma}$ be the fixed point subalgebra of $\mathfrak{g}$, and put $\underline{\mathbf{U}}^-_q = \mathbf{U}^-_q(\mathfrak{g}^{\sigma})$. Let $\mathbf{B}$ be the canonical basis of $\mathbf{U}_q^-$ and $\underline{\mathbf{B}}$ the canonical basis of $\underline{\mathbf{U}}_q^-$. $\sigma$ induces a natural action on $\mathbf{B}$, and we denote by $\mathbf{B}^{\sigma}$ the set of $\sigma$-fixed elements in $\mathbf{B}$. Lusztig proved that there exists a canonical bijection $\mathbf{B}^{\sigma} \simeq \underline{\mathbf{B}}$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.
</p>projecteuclid.org/euclid.jmsj/1572660116_20200115040154Wed, 15 Jan 2020 04:01 ESTGeneralization of Schläfli formula to the volume of a spherically faced simplexhttps://projecteuclid.org/euclid.jmsj/1572660117<strong>Kazuhiko AOMOTO</strong>, <strong>Yoshinori MACHIDA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 37 pages.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to present a variational formula of Schläfli type for the volume of a spherically faced simplex in the Euclidean space. It is described in terms of Cayley–Menger determinants and their differentials involved with hypersphere arrangements. We derive it as a limit of fundamental identities for hypergeometric integrals associated with hypersphere arrangements obtained by the authors in the preceding article.
</p>projecteuclid.org/euclid.jmsj/1572660117_20200115040154Wed, 15 Jan 2020 04:01 ESTAn extension of the characterization of $\mathrm{CMO}$ and its application to compact commutators on Morrey spaceshttps://projecteuclid.org/euclid.jmsj/1572249770<strong>Ryutaro ARAI</strong>, <strong>Eiichi NAKAI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 33 pages.</p><p><strong>Abstract:</strong><br/>
In 1978 Uchiyama gave a proof of the characterization of $\mathrm{CMO}(\mathbb{R}^n)$ which is the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in $\mathrm{BMO}(\mathbb{R}^n)$. We extend the characterization to the closure of $C^{\infty}_{\rm comp}(\mathbb{R}^n)$ in the Campanato space with variable growth condition. As an application we characterize compact commutators $[b,T]$ and $[b,I_{\alpha}]$ on Morrey spaces with variable growth condition, where $T$ is the Calderón–Zygmund singular integral operator, $I_{\alpha}$ is the fractional integral operator and $b$ is a function in the Campanato space with variable growth condition.
</p>projecteuclid.org/euclid.jmsj/1572249770_20200115040154Wed, 15 Jan 2020 04:01 ESTExplicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree levelhttps://projecteuclid.org/euclid.jmsj/1572249776<strong>Martin DICKSON</strong>, <strong>Ameya PITALE</strong>, <strong>Abhishek SAHA</strong>, <strong>Ralf SCHMIDT</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 51 pages.</p><p><strong>Abstract:</strong><br/>
We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.
</p>projecteuclid.org/euclid.jmsj/1572249776_20200115040154Wed, 15 Jan 2020 04:01 ESTStabilities of rough curvature dimension conditionhttps://projecteuclid.org/euclid.jmsj/1571385685<strong>Daisuke KAZUKAWA</strong>, <strong>Ryunosuke OZAWA</strong>, <strong>Norihiko SUZUKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behavior of metric measure spaces satisfying the rough curvature dimension condition. We prove stabilities of the rough curvature dimension condition with respect to the observable distance function and the $L^2$-transportation distance function.
</p>projecteuclid.org/euclid.jmsj/1571385685_20200115040154Wed, 15 Jan 2020 04:01 ESTApéry–Fermi pencil of $K3$-surfaces and 2-isogenieshttps://projecteuclid.org/euclid.jmsj/1571212902<strong>Marie José BERTIN</strong>, <strong>Odile LECACHEUX</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 39 pages.</p><p><strong>Abstract:</strong><br/>
Given a generic $K3$-surface $Y_k$ of the Apéry–Fermi pencil, we use the Kneser–Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda–Inose structure. The other fibrations correspond to a $K3$-surface identified by its transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations between 2-isogenies and isometries on the singular surfaces of the family.
</p>projecteuclid.org/euclid.jmsj/1571212902_20200115040154Wed, 15 Jan 2020 04:01 ESTAsymptotic behavior of eigenfrequencies of a thin elastic rod with non-uniform cross-sectionhttps://projecteuclid.org/euclid.jmsj/1571212903<strong>Shuichi JIMBO</strong>, <strong>Albert Rodríguez MULET</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 36 pages.</p><p><strong>Abstract:</strong><br/>
We study the eigenvalue problem of the elliptic operator which arises in the linearized model of the periodic oscillations of a homogeneous and isotropic elastic body. The square of the frequency agrees to the eigenvalue. Particularly, we deal with a thin rod with non-uniform connected cross-section in several cases of boundary conditions. We see that there appear many small eigenvalues which accumulate to $0$ as the thinness parameter $\varepsilon$ tends to $0$. These eigenvalues correspond to the bending mode of vibrations of the thin body. We investigate the asymptotic behavior of these eigenvalues and obtain a characterization formula of the limit equation for $\varepsilon \rightarrow 0$.
</p>projecteuclid.org/euclid.jmsj/1571212903_20200115040154Wed, 15 Jan 2020 04:01 ESTSmall gaps between the set of products of at most two primeshttps://projecteuclid.org/euclid.jmsj/1566871231<strong>Keiju SONO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 38 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we apply the methods of Maynard and Tao to the set of products of two distinct primes ($E_{2}$-numbers). We obtain several results on the distribution of $E_{2}$-numbers and primes. Among others, the result of Goldston, Graham, Pintz and Yıldırım on small gaps between $m$ consecutive $E_{2}$-numbers is improved.
</p>projecteuclid.org/euclid.jmsj/1566871231_20200115040154Wed, 15 Jan 2020 04:01 ESTDynamics of isolated left ordershttps://projecteuclid.org/euclid.jmsj/1566871232<strong>Shigenori MATSUMOTO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
A left order of a countable group $G$ is called isolated if it is an isolated point in the compact space $LO(G)$ of all the left orders of $G$. We study properties of a dynamical realization of an isolated left order. Especially we show that it acts on $\mathbb{R}$ cocompactly. As an application, we give a dynamical proof of the Tararin theorem which characterizes those countable groups which admit only finitely many left orders. We also show that the braid group $B_3$ admits countably many isolated left orders which are not the automorphic images of the others.
</p>projecteuclid.org/euclid.jmsj/1566871232_20200115040154Wed, 15 Jan 2020 04:01 ESTDefinability of singular integral operators on Morrey–Banach spaceshttps://projecteuclid.org/euclid.jmsj/1563501704<strong>Kwok-Pun HO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We give a definition of singular integral operators on Morrey–Banach spaces which include Orlicz–Morrey spaces and Morrey spaces with variable exponents. The main result of this paper ensures that the singular integral operator is well-defined on the Morrey–Banach spaces. Therefore, it provides a solid foundation for the study of singular integral operators on Morrey type spaces. As an application of our main result, we study commutators of singular integral operators on Morrey–Banach spaces.
</p>projecteuclid.org/euclid.jmsj/1563501704_20200115040154Wed, 15 Jan 2020 04:01 ESTSome modules over Lie algebras related to the Virasoro algebrahttps://projecteuclid.org/euclid.jmsj/1562033193<strong>Guobo CHEN</strong>, <strong>Jianzhi HAN</strong>, <strong>Yucai SU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study restricted modules over a class of $(1/2) \mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ related to the Virasoro algebra. We in fact give the classification of certain irreducible restricted $\mathfrak{g}$-modules in the sense of determining each irreducible restricted module up to an irreducible module over a subalgebra of $\mathfrak{g}$ which contains its positive part. Several characterizations of these irreducible $\mathfrak{g}$-modules are given. By the correspondence between restricted modules over $\mathfrak{g}$ and modules over the vertex algebra associated to $\mathfrak{g}$, we get the classification of certain irreducible modules over vertex algebras associated to these $\mathfrak{g}$.
</p>projecteuclid.org/euclid.jmsj/1562033193_20200115040154Wed, 15 Jan 2020 04:01 ESTThe strong slope conjecture and torus knotshttps://projecteuclid.org/euclid.jmsj/1558425754<strong>Efstratia KALFAGIANNI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/>
We show that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a $(2,q)$-torus knot.
</p>projecteuclid.org/euclid.jmsj/1558425754_20200115040154Wed, 15 Jan 2020 04:01 ESTCocycles of nilpotent quotients of free groupshttps://projecteuclid.org/euclid.jmsj/1552896022<strong>Takefumi NOSAKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We focus on the cohomology of the $k$-th nilpotent quotient of a free group. We describe all the group 2-, 3-cocycles in terms of the Massey product and give expressions for some of the 3-cocycles. We also give simple proofs of some of the results on the Milnor invariant and Johnson–Morita homomorphisms.
</p>projecteuclid.org/euclid.jmsj/1552896022_20200115040154Wed, 15 Jan 2020 04:01 ESTSome infinitely generated non-projective modules over path algebras and their extensions under Martin's axiomhttps://projecteuclid.org/euclid.jmsj/1579165217<strong>Ayako ITABA</strong>, <strong>Diego A. MEJÍA</strong>, <strong>Teruyuki YORIOKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X}, KQ) = 0$ then $\mathcal{X}$ is projective. In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ such that, when $K$ is a countable field, $\mathbf{MA}_{\aleph_1}$ (Martin's axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1}, KQ) = 0$.
</p>projecteuclid.org/euclid.jmsj/1579165217_20200116040049Thu, 16 Jan 2020 04:00 EST$\mu$-type subgroups of $J_1(N)$ and application to cyclotomic fieldshttps://projecteuclid.org/euclid.jmsj/1582167611<strong>Masami OHTA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 72, Number 2, 333--412.</p><p><strong>Abstract:</strong><br/>
Let $p$ be an odd prime number, and $N$ a positive integer prime to $p$. We prove that $\mu$-type subgroups of the modular Jacobian variety $J_1(N)$ or $J_1(Np)$ of order a power of $p$ and defined over some abelian extensions of $\mathbb{Q}$ are trivial, under several hypotheses. For the proof, we use the method of Vatsal. As application, we show that a conjecture of Sharifi is valid in some cases.
</p>projecteuclid.org/euclid.jmsj/1582167611_20200422220158Wed, 22 Apr 2020 22:01 EDTOn Hamiltonian stable Lagrangian tori in complex hyperbolic spaceshttps://projecteuclid.org/euclid.jmsj/1580893211<strong>Toru KAJIGAYA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 72, Number 2, 435--463.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n \leq 2$ and there exist infinitely many H-unstable $T^n$-orbits when $n \geq 3$. On the other hand, we prove a monotone $T^n$-orbit in $\mathbb{C}H^n$ is H-stable and rigid for any $n$. Moreover, we see almost all Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ are not Hamiltonian volume minimizing when $n \geq 3$ as well as the case of $\mathbb{C}^n$ and $\mathbb{C}P^n$.
</p>projecteuclid.org/euclid.jmsj/1580893211_20200422220158Wed, 22 Apr 2020 22:01 EDTDel Pezzo surfaces with a single $1/k(1,1)$ singularityhttps://projecteuclid.org/euclid.jmsj/1581498014<strong>Daniel CAVEY</strong>, <strong>Thomas PRINCE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 72, Number 2, 465--505.</p><p><strong>Abstract:</strong><br/>
Inspired by the recent progress by Coates–Corti–Kasprzyk et al. on mirror symmetry for del Pezzo surfaces, we show that for any positive integer $k$ the deformation families of del Pezzo surfaces with a single $1/k(1,1)$ singularity (and no other singular points) fit into a single cascade. Additionally we construct models and toric degenerations of these surfaces embedded in toric varieties in codimension $\leq 2$. Several of these directly generalise constructions of Reid–Suzuki (in the case $k = 3$). We identify a root system in the Picard lattice, and in light of the work of Gross–Hacking–Keel, comment on mirror symmetry for each of these surfaces. Finally we classify all del Pezzo surfaces with certain combinations of $1/k(1,1)$ singularities for $k = 3,5,6$ which admit a toric degeneration.
</p>projecteuclid.org/euclid.jmsj/1581498014_20200422220158Wed, 22 Apr 2020 22:01 EDT