In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

The complement of an arrangement of hyperplanes is a good example of the mixed Hodge structure on the fundamental group of an algebraic variety. We compute its isomorphic class using iterated integrals in the fiber type case and then get the combinatorial and projective invariant.

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

Let S = K[x1,x2,…,xn] be a polynomial ring over a field K. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integer k ≥ 0, we investigate the k-Buchsbaum property of residue class rings S/I(t); and S/It for the Stanley-Reisner ideal I = IΔ. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex Δ and the power t. We also give a characterization in the case where I is the edge ideal of a simple graph.

In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

Let A be a real analytic set, M be a compact real analytic manifold and f : A × M → R be a real analytic function. Then we have a family of real analytic functions fa, a ∈ A, on M defined by fa(X) = f(a, x).

A density P on the punctured unit disk Ω:0 < |z| <1 is a 2-form P(z)dxdy whose coefficient P(z) is a real valued nonnegative locally Hölder continuous function on the closed punctured unit disk Ω:0< |z| <≦1. Here we consider Ω as an end of the punctured sphere 0 < |z| ≦ + + so that the point z = 0 is viewed as the ideal boundary δΣ of Σ and the unit circle |z| = 1 as the relative boundary δΣ of Σ. We denote by D = D(Σ) the family of densities on Σ.

We study the Fourier coefficients of cusp forms of half integral weight and generalize the Kohnen-Zagier formula to the case of ‘4 × general odd’ level by using results of Ueda. As an application, we obtain a generalization of the result of Luo-Ramakrishnan [11] to the case of arbitrary odd level.

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a nonarchimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin–Selberg $L$-factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$ is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$. As an application we prove that the $L$-factor of a pair of banal $\ell$-modular cuspidal representations is the reduction modulo $\ell$ of the $L$-factor of any pair of $\ell$-adic lifts.

In this paper, we prove that the set of equivalence classes of dormant opers of rank $p-1$ over a projective smooth curve of genus ${\geqslant}2$ over an algebraically closed field of characteristic $p>0$ is of cardinality one.

In connection with Serre’s problem, we consider a locally trivial analytic fibration π : E → B of complex spaces with typical fiber X. We show that if X is a Stein curve and B is q-complete, then E is q-complete.

All rings in this paper are assumed to be commutative with identity and the terminology is standard.

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

Utilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $\mathcal {B}(R)$ over a local domain R essentially of finite type over $\mathbb {C}$. We show that if R is normal and $\Delta $ is an effective $\mathbb {Q}$-Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$-Cartier, then the BCM test ideal $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ coincides with the multiplier ideal $\mathcal {J}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$, where $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$-adic completions of R and $\mathcal {B}(R)$, respectively, and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$. As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.

Let K be an extension of a field k, and p denotes the characteristic. It was proved by M. Gerstenhaber ([1]) that if K is separable over k, then it is rigid and it was conjectured in [1] that, if K is not separable over k, then it is not rigid. We studied in [4] the above conjecture in certain special case. In this note we shall extend the results of [4] to inseparable algebraic extension fields.

In the present paper by an algebraic system (algebra) A we shall mean a system with a set F of operations fλ: (x1,…, xn) ∈ A × · · · × A → fλ(x1,…, xn) ∈ A. A polynomial p(x1, …, xr) is a function of variables x1,…, xr which is either one of the xi, or (recursively) a result of some operation fλ(p1,…, pn) performed on other polynomials pi. An algebra A may satisfy a set R of identities p(x1,…, xr) = q(x1,…, xs), and then A shall be called an (F, R)-algebra.

Let M > 1 be a positive number. Let be a family of holomorphic functions f in some domain D ⊂ ℂ for which there exists an integer k = k(f) > 2 such that |(fk)′(ζ)| ≤ Mk for every periodic point ζ of period k of f in D. We show first that is quasinormal of order at most one in D. This strengthens a result of W. Bergweiler. Secondly, for the case M = 1, we prove that is normal in D if there exists a positive number K < 3 such that | f(η)| ≤ K for each f ∈ and every fixed point η of f in D. This improves a result of M. Esséen and S. J. Wu. We also construct an example which shows that the condition |f’(η)| ≤ K < 3 can not be replaced by | f′(η) | < 3.

Let X be a 3-dimensional terminal singularity of index ≥ 2. We study projective birational morphisms ϕ: Y → X such that the exceptional divisor of ϕ consists of all prime divisors with discrepancies < 1 (resp. ≤ 1) over X.

Let be a Kac-Moody-Borcherds algebra on a field associated to a symetrizable matrix and with Cartan subalgebra . Let be an ad -invariant subalgebra such that the restriction to of the standard bilinear form is nondegenerate. We show that the root system Ψ of is a subsystem according to [Ba] of . Moreover, if a subsystem Ω satisfies some conditions (i.e. Ω is “réduit et presque-clos”) of Ψ, we construct inside of a Kac-Moody-Borcherds algebra with root system Ω.

Let k be a subfield of . We prove similar results in the case of an action of a finite group of k-semi-automorphisms. In particular, we obtain a generalization to the Kac-Moody case of a result by Borel and Tits.

Let be an almost-k-split form of a Kac-Moody algebra. We construct a Kac-Moody k-algebra with root system similar to the system of (save on some multiples of certain roots).

It is known classically that abelian varieties of dimension one over the field of complex numbers may be expressed by non-singular Hesse’s canonical cubic plane curves, The purpose of the present paper is to generalize this idea to higher dimensional case.