Let SF = SG denote the space , and BSF be the classifying space of SF. Our purpose is to determine H*(BSF: Zp) as a Hopf algebra over Zp where p is an odd prime number. We have announced the main result in [14].

Let k be an algebraically closed field of characteristic p≧ 0, and let X be an abelian variety over k.

The goal of this paper is to answer the following questions, when dim(X) = 1 and p≠0, posed by R. Hartshorne:

• (1) Is E(P) indecomposable, when E is an indecomposable vector bundle on X?

• (2) Is the Frobenius map F*: H1 (X, E) → H1 (X, E(p)) injective?

  We also partly answer the following question posed by D. Mumford:

• (3) Classify, or at least say anything about, vector bundles on X when dim (X) > 1.

In our earlier paper [4] we developed the basic sheaftheoretic and cohomological properties of ample vector bundles. These generalize the corresponding well-known results for ample line bundles. The numerical properties of ample vector bundles are still poorly understood. For line bundles, Nakai’s criterion characterizes ampleness by the positivity of certain intersection numbers of the associated divisor with subvarieties of the ambient variety. For vector bundles, one would like to characterize ampleness by the numerical positivity of the Chern classes of the bundle (and perhaps of its restrictions to subvarieties and their quotients). Such a result, like the Riemann-Roch theorem, giving an equivalence between cohomological and numerical properties of a vector bundle, may be quite subtle. Some progress has been made by Gieseker [2], by Kleiman [8], and in the analytic case, by Griffiths [3].

In [9], Hartshorne extended the concept of ampleness from line bundles to vector bundles. At that time, he conjectured that the appropriate Chern classes of an ample vector bundle were positive, and it was hoped that there would be some criterion for ampleness of vector bundles similar to Nakai’s criterion for line bundles. In the same paper, Hartshorne also introduced the notion of p-ample when the ground field had characteristic p, proved that a p-ample bundle was ample and asked if the converse were true.

Let M be an oriented hypersurface differentiably immersed in a Euclidean space of n + 1 ≥: 3 dimensions. The r-th mean curvature Kr of M at the point P of M is defined by the following equation:

The cohomology theory of an associative algebra has been shown to be valuable in the study of the structure of algebras of finite cohomological dimension, especially those of dimension less than or equal to one over a field. M. Harada [9] has shown that every semi-primary hereditary algebra A (for example, A is finitely generated over a field R and has dimension < 1) is isomorphic to a generalized triangular matrix algebra. The concept of a central separable algebra over a commutative ring has been shown to be a useful generalization of the concept of a central simple algebra over a field, where a separable algebra is defined to be an algebra having cohomological dimension zero.

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.

Let k be a p-adic number field and o be the ring of all integers in k. Moreover, let K\k be a finite Galois extension with the Galois group G = G(K/k). Then the ring of all integers in K is an -module.

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote by Ω the Riemann sphere. If f(z) is a meromorphic function in D, and if ζ∈Г, then the principal cluster set of f at ζ is the set

1. Fatou’s theorem is true for a bounded complex-valued harmonic function in the disk D: |z|<1. One asks naturally: “Is Meier’s topological analogue of Fatou’s theorem (simply, “MF theorem”; [14, p. 330, Theorem 6], cf. [10, p. 154, Theorem 8.9]) true for a bounded complex-valued harmonic function in D?” We shall give the affirmative answer to this question. Furthermore, the horocyclic MF theorem [2, p. 14, Theorem 5] in the complex-harmonic form will be proved in parallel.

Let f be an arbitrary function from the open unit disk D of the complex plane into the Riemann sphere S. If p is any point on the unit circle C, C(f, p) is the set of all points w such that there exists in D a sequence of points {Zj} such that zj→p and f(zj)→w. CΔ(f, p) is defined in the same way, but the sequence {Zj} is restricted to Δ⊂D. If α and β are two arcs in D terminating at p and Cα(f, p)∩Cβ(f, p) = Φ, p is called an ambiguous point for f.

Let BSPL be the classifying space of the stable oriented PL micro bundles. The purpose of this paper is to determine H*(BSPL:Zp) as a Hopf algebra over Zp, where p is an odd prime number. In this chapter, p is always an odd prime number.

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that