As was shown by a part of the authors, for a given $(2,3,5)$-distribution $D$ on a five-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another five-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y,D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2,3,5)$-distributions. Thus, we complete the duality between $(2,3,5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_{2}$. A local example of nonflat perturbations of the global model of flat Lagrangian cone structure which corresponds to $(2,3,5)$-distributions is given.

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

In this paper, we construct Chern classes from the relative $K$-theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.

We give a presentation of cyclotomic q-Schur algebras by generators and defining relations. As an application, we give an algorithm for computing decomposition numbers of cyclotomic q-Schur algebras.

D. Mumford has shown an excelent algebralization of theory of theta constants and theta functions in his papers: On the equations defining abelian varieties I, II, III (Invent. Math. 1. 237-354 (1966), 3. 75-135 (1967), 3. 215-244) (1967). Our starting point and idea, however, are something different from those of Mumford; we begin our study at characterizing abelian addition formulae among all the possible addition formulae, and we want to give expressions to everything in words of matric notations.

Let k be a -adic number field with the ring 0 of all integers in k, and K be a finite normal extension with Galois group G. ∏ denotes a prime element of the ring of all integers in K. Then, an ideal (∏a) of is an 0G-module. E. Noether showed that if K/k is tamely ramified, is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows: is relatively projective with respect to a subgroup S of G if and only if S ⊇ G1 where G1 is the first ramification group of K/k. Now we define the vertex V(∏a) of (∏a) as the minimal normal subgroup S of G such that (∏a) is relatively projective with respect to a subgroup S of G (cf. § 1). Then, the above generalization by A. Fröhlich implies V() = G1. In the previous paper, we proved G1 ⊇ V(∏a) ⊇ G2, (where G2 is the second ramification group of K/k (cf. Theorem 5). Further, we dealt with the case where G = G1 is of order p2, and proved that if V(∏a) ≠ G1 then a ≡ 1(p2) and t2 ≡ 1(p2) for the second ramification number t2 of K/k (cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degree pn (Theorem 7).

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand et al. [Generalized Euler integrals and$A$-hypergeometric functions, Adv. Math. 84 (1990), 255–271] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves and twisted Morse theory are used.

N. Aronszajn introduced in [4] an abstract Frechét space R (0<R≤∞), which is isomorphic to the space of analytic solutions of the heat equation in if 0 < R ∞, and in if R = ∞, and called it the space of traces of analytic solutions of the heat equation. Hereafter, we call it the space of traces, shortly.

Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Let Ω be the projective limit space of a sequence of probability space Ωn which is a certain subset of (n − 1)-dimensional sphere with the usual uniform probability distribution on it. T. Hida [2], starting from a sequence of finite dimensional flows which are derived from some one-parameter subgroups of rotations of spheres, constructed a flow {Tt} on Ω as the limit of them. Observing his method, the concept of consistency of flows which approximate {Tt} seems to play an essential role in his work [2]. As will be made clear in the following sections, the concept of consistency is closely related to the projective limiting structure of our basic space Ω. The purpose of this paper is to determine all the flows on Ω which can be approximated in the sense of [2] by finite dimensional flows.

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.

In the previous paper [3], we introduced new coordinates to the Schottky space, and defined the augmented Schottky spaces . Here, in § 1, we will define fiber spaces over the augmented Schottky spaces. In §2, we will consider a property of the new coordinates, namely, we will state a relation between limits of sequences of elements of the new coordinates and limits of sequences of length of loops on Riemann surfaces.

An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g) and the linear subspace ℙ*(g1) in ℙ*(g) coincides with the cubic Veronese variety associated to g.

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

Let f: E→ P′(C) be an elliptic fibration with non-constant j-invariant, and possessing a section σ0. The group of sections of f is then naturally identified with the group of points defined over a function field C(u) of the generic fibre of f.

Recently H. Hasse has given an interesting theory of Galois algebras, which generalizes the well known theory of Kummer fields; an algebra over a field Ω is called a Galois algebra with Galois group G when possesses G as a group of automorphisms and is (G, Ω)-operator-isomorphic to the group ring G(Ω) of G over Ω. On assuming that the characteristic of Ω does not divide the order of G and that absolutely irreducible representations of G lie in Ω, Hasse constructs certain Ω-basis of , called factor basis, in accord with Wedderburn decomposition of the group ring and shows that a characterization of is given by a certain matrix factor system which defines the multiplication between different parts of the factor basis belonging to different characters of G. Now the present work is to free the theory from the restriction on the characteristic. We can indeed embrace the case of non-semisimple modular group ring G(Ω).