Showing papers in "Osaka Journal of Mathematics in 1973"

On homogeneous real hypersurfaces in a complex projective space

Abstract: The purpose of this paper is to determine those homogeneous real hypersurfaces in a complex projective space Pn(C) of complex dimension n(^>2) which are orbits under analytic subgroups of the projective unitary group PU(n-\\-\\)> and to give some characterizations of those hypersurfaces. In § 1 from each effective Hermitian orthogonal symmetric Lie algebra of rank two we construct an example of homogeneous real hypersurface in Pn(C)y which we shall call a model space in Pn(C). In §2 we show that the class of all homogeneous real hypersurfaces in Pn{C) that are orbits under analytic subgroups of PU(n-\\-l) is exhausted by all model spaces. In §§3 and 4 we give some conditions for a real hypersurface in Pn(C) to be an orbit under an analytic subgroup of PU(n-\\-l) and in the course of proof we obtain a rigidity theorem in Pn(C) analogous to one for hypersurfaces in a real space form. The author would like to express his hearty thanks to Professor T. Takahashi for valuable discussions with him and his constant encouragement, and to Professor M. Takeuchi who made an original complicated proof of Lemma 2.3 short and clear.

Elliptic curves of prime power conductor with q-rational points of finite order

Abstract: Let E be an elliptic curve (an abelian variety of dimension one) defined over the rational number field Q. After Weil [9], we can define the conductor of E. But, in general, it would be difficult to find all the curves of given conductor. But it seems to be easier to find all the curves of given conductor having Q-rational points of finite order >2. In this paper we determine all the curves of prime power conductor which have at least three rational points of finite order. There are only finitely many such curves up to Q-isomorphism. They are listed in the table at the end of the paper. Since each of them has Q-rational points of finite order, we can take a special cubic equation as a global minimal model for it. Further, since that curve has a prime power conductor, the coefficients of that equation must be a solution of a certain diophantine equation. Therefore, the determination of such curves is reduced to elementary diophantine problems. Some of them have no complex multiplication and non-integral invariants. Let E be one of them and Q(En) be the field generated by the coordinates of the ^-division points of E over Q. Then we can determine the Galois group of Q{El) over Q for all prime /.

A geometric meaning of the multiplicity of integrable discrete classes in L²(Γ＼G)

Abstract: Let F be a discrete subgroup of a group G of motions of a noncompact symmetric space M of inner type such that the quotient T\M is compact. Let K be the isotropy group at a point of M. Let co be a discrete class of G such that infinite matrix coefficients of co belong to ^(G). Depending on some parameter corresponding to co we can associate a homogeneous vector bundle over M. The Dirac operator D which is a first order elliptic G-invariant differential operator acts on the space of C°° sections of the above vector bundle. We then prove (Theorem 4, §3) that the multiplicity NJJ*) of co in the (right) regular representation of G on L(T\G) is the dimension of the space H(T; *) of sections which are annihilated by D (Dirac spinors) and which are F-invariant and obtain a formula for the same (Corollary to Theorem 4, §3). We remark that algebraic formulas for iVw(r) are already available in several cases (Langlands [6]). Also when the parameter corresponding to co satisfies some further conditions, Schmid [9] obtained geometric meaning to the multiplicity NJT) (similar to Theorem 4), working with G\T rather than the symmetric space GjK where T is a Cartan subgroup of G contained in K. Our method of proof is as follows: The space H(T; *) is the direct sum of two subspaces H(T; *) and H~(T; *). First we prove that one of these two spaces vanishes (Theorem 2, §1). Then using the Lefschetz Theorem of Atiyah and Singer [1] we obtain a formula for the difference dim H(T; *) — dim H~(T; *) (Theorem 3, §2). Let us divide our problem into two parts; namely, (1) to prove dim H(T; *)=j\Ttt(r) and (2) to compute explicitly the above number. When F has no elliptic elements other than the identity, we prove (1) by directly showing** that the expression for dim H(T; #) — dim H~(T; *) given by Theorem

On homogeneous Kähler manifolds with non-degenerate canonical Hermitian form of signature (2, ,2(n-1))

Abstract: We denote by M a connected homogeneous Kahler manifold of complex dimension n on which a connected Lie group G acts effectively as a group of holomorphic isometries, and by K an isotropy subgroup of G at a point o of M. Let v be the G-invariant volume element corresponding to the Kahler metric. In a local coordinate system {z19 •••,#„}, v has an expression v=i nFdz1Λ Ά . . . . g logF dznΛd21Λ\"-Λd2n. The G-invaπant hermitian form h= 2 —dz4dSj is

Multiplicative $P$-subgroups of simple algebras

Abstract: Amitsur ([1]) determined all finite multiplicative subgroups of division algebras. We will try to determine, more generally, multiplicative subgroups of simple algebras. In this paper we will characterize />-groups contained in full matrix algebras MM(A) of fixed degree ft, where A are division algebras of characteristic 0. All division algebras considered in this paper will be of characteristic 0. Let A be a division algebra. We will denote by Mn(A) the full matrix algebra of degree n over A. By a subgroup of MM(A) we will mean a multiplicative subgroup of Mn( A). Further let K be a subfield of the center of A and let G be a finite subgroup of MM(A). Now we define VK(G)={^2aigi\\ai^Ky g{^G}. Then VK(G) is clearly a J^-subalgebra of Mn(A) and there is a natural epimorphism KG->VK(G) where KG denotes the group algebra of G over K. Hence VK(G) is a semi-simple i^-subalgebra of Mn(A), which is a direct summand of KG. As usual Qy Ry C, H denote respectively the rational number field, the real number field, the complex number field and the quaternion algebra over R. If an abelian group G has invariants (ely •••, en), en^=l, ei+1 \\ eiy we say briefly that G has invariants of length ft. We begin with

Modules over Dedekind prime rings. V

Abstract: The purpose of this paper is the investigation of modules over Dedekind prime rings. In Section 1, we shall prove that the double centralizer of a P-primary module over a Dedekind prime ring R is isomorphic to KP or ήP/P , where P is a nonzero prime ideal of R and jfeP is the P-adic completion of R with unique maximal ideal P. Using this result we shall determine the structure of the double centralizer of primary modules over bounded Dedekind prime rings. In Section 2, we shall give a characterization of quasi-injective modules over bounded Dedekind prime rings. This paper is a continuation of [7] and [8]. A number of concepts and results are needed from [7] and [8].