In this paper, we shall investigate the hypoellipticity for a class of degenerate equations of the second order with complex coefficients as a direct extension of the results obtained in [8]. As is well known, the satisfactory general results about hypoellipticity of real operators of the second order have been obtained in [3] and [9], where the assumption that the operators are real plays a crucial role and our aim of this paper is to study the operators with complex coefficients. Our method may be considered as a generalization of the usual variational method replacing the Gårding inequality by the estimate (2.15), (cf. [3], [5]).

Let Ω be a bounded domain in ndimensional Euclidian space Rn(n ≧ 2)with C∞boundary Γ of dimension n— 1 and let there be given two real-valued C∞–functions α, β on Γ such that α≧ 0, β ≧ 0and α + β = lthroughout Γ.Then we consider the non-elliptic boundary value problem with λ ≧ 0 (which is always assumed, and in particular when λ =0, we further assume β ≢ 0,throughout this paper) :

(1)

Let us consider the stochastic differential equation,

with initial condition,

where Bt, t ≧ 0,is a one-dimensional Brownian motion, and Lxis a second order uniformly elliptic partial differential operator satisfying some additional conditions that will be described in §2. The existence and the uniqueness of solutions of the Cauchy problem have been established by B. L. Rozovskii [8].

Soit un système transcendant dans le plan c’est-à-dire, les fonctions f0,...,fnsont entières sans zéros communs à toutes et

Dans toute la suite Xdésignera un groupe abélien localement compact et séparé, et ξdésignera la mesure de Haar sur X.Pour simplifier la discussion, on supposera toujours que Xest à base dénombrable.

In the previous paper [1], we showed that the set of simple vector bundles of rank 2 on a rational surface with fixed Chern classes is bounded and we gave a sufficient condition for an H-stable vector bundle of rank 2 on a rational surface to be ample. In this paper, we shall extend the results of [1] to the case of higher rank.

Let k be an algebraic number field, let K be a Galois extension of k of finite degree, and let OK, Ok be the maximal orders of K, k, respectively. We consider the conjugate operation: for a given quadratic lattice M over OK equipped with a bilinear form B and for an automorphism σ ∈ G(K/k), we define a new quadratic lattice Mσ over OK. Here Mσ has the same underlying abelian group as M, but a new OK-action ; the new bilinear form Bσ on Mσ is defined by

Let E be an elliptic curve defined over the field Q of rational numbers, then the torsion subgroup of the Mordell-Weil group E(Q) is finite and it is known that there exist the elliptic curves whose torsion subgroups E(Q)t are of the following types: (1), (2), (3), (2, 2), (4), (5), (2, 3), (7), (2, 4), (8), (9), (2, 5), (2, 2, 3), (3, 4) and (2, 8). It has been conjectured from various reasons that E(Q)t is exhausted by the above types only. If E has a torsion point of order precisely n, then it is known that E has an n-isogeny, that is to say, an isogeny of degree n.

Let Sbe a compact complex protective and smooth variety of dimc2, shortly a surface. We employ the following standard notations:

Os: structure sheaf of S.

Ks: the canonical bundle on S.

The following question was the main motivation of this paper : which are the necessary and sufficient conditions for a non-singular subvariety of a Grassmann variety to have an ample normal bundle. Knowing that a non-singular subvariety of a Grassmann variety has an ample normal bundle we can apply on it several well-known theorems.

Let E/F be a finite extension of algebraic number fields, OE, OF the maximal orders of E, F respectively. A classical theorem of Springer [6] asserts that an anisotropic quadratic space over F remains anisotropic over E if the degree [E: F] is odd. From this follows that regular quadratic spaces U, V over F are isometric if they are isometric over E and [E : F] is odd. Earnest and Hsia treated similar problems for the spinor genera [2, 3]. We are concerned with the quadratic lattices. Let L, M be quadratic lattices over OF in regular quadratic spaces U, V over F respectively.

In this paper, we shall discuss boundedness of solutions of the equation

under suitable conditions. And we shall discuss asymptotic stability of a periodic solution and convergence of solutions for the equation

for a positive constant cand a periodic function e(t)under some restricted conditions.

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g, be a congruent ideal class group of K, and M be the class field over K corresponding to . Assume that M is normal over k. Then g acts on as a group of automorphisms. Donote by lg the augmentation ideal of the group ring Zg over the ring of integers Z.

In [1], G. Shimura gave a very practical formula of the traces of the Hecke operators acting on the space of cusp forms of rational weight and there he emphasized that the traces are effectively computable. We shall practice the computation in some special cases and discuss the structure of the Hecke algebra, which is not necessarily semi-simple.