The trace of Hecke operators with respect to a unit group of an order in a quaternion algebra has been given in Eichler [1], [2] in the case when the order is of square-free level. The purpose of this note is to study the order of type (q1, q2, q3) (see text 1.1), in the case, of cube-free level, and to give a formula for the trace of Hecke operators in the case q3 = 2.

A connected complex Lie group without non-constant holomorphic functions is called a toroidal group ([5]) or an (H, C)-group ([9]). Let X be an n-dimensional toroidal group. Since a toroidal group is commutative ([5], [9] and [10]), X is isomorphic to the quotient group Cn/Γ by a lattice of Cn. A complex torus is a compact toroidal group. Cousin first studied a non-compact toroidal group ([2]).

Let L be a holomorphic line bundle over X. L is said to be homogeneous if is isomorphic to L for all x ε X, where Tx is the translation defined by x ε X. It is well-known that if X is a complex torus, then the following assertions are equivalent:

In this paper we are concerned with transversely orientable codimension one foliations. Let be a Cr-foliation as above in a smooth manifold M, r ≧ 1, and let F0 be a closed leaf of . A neighborhood U of F0 is called a bicollar of F0 in this paper if there is a normal line bundle : U → F0 with respect to a fixed Riemannian metric on M such that each fibre of is transverse to .

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.

1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;

(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.

Let Rn be the Euclidean space of dimension n ≧ 1 with the standard inner product and the norm be the unit sphere {x ∈ Rn; |x| = 1} and dωn-1 be the volume element of Sn-1 such that Sn-1 gets the volume 1. Let Ω be an open set of Rn containing Sn-1 and let f: Ω → Rm be a smooth map.

Consider a one-dimensional differential algebraic function field K over an algebraically closed ordinary differential field k of characteristic 0. We shall prove the following theorem:

Suppose that the group of all automorphisms of K over k is infinite. Then, K is either a differential elliptic function field over k or K = k(ν) with ν′ = ξ or ν′ = ην, where ξ, η ϵ k.

This paper is concerned with the cohomology at “infinity” (in the sense of Harder [4], [5]) of a congruence subgroup of the symplectic group G = Sp(2l, R). G is the subgroup of GL(2l, R) consisting of matrices g satisfying tgJg = J where

We prove that the maximal number of conics in a smooth sextic $K3$ -surface $X\subset \mathbb {P}^4$ is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible.

In the theory of modular forms there is an interesting problem whether every modular form can be expressed as a linear comination of theta functions. For this Eichler proved in [1] that for a sufficiently large prime q all modular forms of degree — 2m(m = 1,2, · · ·) for Γ0(q) can be represented by linear combinations of theta functios of degree — 2m with level 1 and q. We prove this theorem for q = 2, 3, 5 and 11 by using a theorem of Siegel for q = 2, 3, 5 and a general result of Eichler for q = 11. The former method is shown in Schoeneberg [2].

In this paper, I would like to point out some characteristic features of formal systems having just one primitive notion. Most remarkable systems of this kind may be Zermelo’s and Fraenkel’s set-theories, both having just one primitive notion ∈.

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totally

geodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .

In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell }$, with $\ell$ different from $p$. We define “nilpotent lifts” of irreducible generic $k$-representations of $GL_{n}(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$-representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$. This gives a characterization of the mod-$\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

Let P(x, D) be a linear partial differential operator with real analytic coefficients and let C ⊂ Rn be a germ of closed subset, say at the origin. We say that C is (the locus of) an irremovable singularity of a real analytic solution u of P(x, D)u = 0 if u is defined outside C on a neighborhood Ω of 0 but cannot be extended to the whole neighborhood Ω even as a hyperfunction solution of P(x, D)u = 0. This usage of the word “singularity” is the same as the one for the analytic functions in complex analysis, and is different of the usual usage of “singularities of solutions” in the theory of partial differential equations.

In 1969, Mumford [8] proved that, for a complete non-singular algebraic surface F over the complex number field C, the dimension of the Chow group of zero-cycles on F is infinite if the geometric genus of F is positive. To this end, he defined a regular 2-form ηf on a non-singular variety S for a regular 2-form η on F and for a morphism f: S → SnF, where SnF is the 72-th symmetric product of F, and he showed that ηf vanishes if all 0-cycles f(s), s ∈ S, are rationally equivalent. Roitman [9] later generalized this to a higher dimensional smooth projective variety V.

Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equation

where ▽u denotes the Laplacian of u, defined with respect to the metric g.

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)

In this note, we treat the problem to determine the conformal structure of the closed surface by the structure of the differentiable function algebra as the normed algebra with a certain norm.

A similar investigation is found in Myers [1]. He concerns himself with determining the Riemannian structure of the compact manifold using a certain normed algebra of differentiable functions.

Beurling and Deny [1], [2] introduced the notion of Dirichlet spaces. They [2] showed the existence of balayaged measures and equilibrium measures in the theory of Dirichlet spaces. In this paper, we shall show that the following equivalence is valid for a Dirichlet space on a locally compact Hausdorff space X.