In our previous paper [3] we proved the following ([3, Theorem 16]) :

THEOREM A. Let G be a 2-group of class 3. Let G2 and G/G2 be direct products of cyclic groups 〈yq〉 of order αq (1 ≦ q ≦ m), and of cyclic groups 〈hi〉 of order βi (1 ≦ i ≦ n) with β1 ≧ β2 ≧ · · · βn, respectively. Let xi be representatives of hi (1 ≦ i ≦ n), and put Then a homomorphism ψ:G3→T can be extended to a polynomial map from G to T of degree ≦ 4 if and only if there exists an integral solution in the following linear equations of Xiq (1 ≦ i ≦ n, 1 ≦ q ≦ m) with coefficients in T:

(I)

where δij is the Kronecker symbol for βi: i.e. δij = 1 or 0 according to βi = βj or βi > βj, respectively.

This is the continuation of our previous paper [3], and will complete, without homogeneity assumption, the classification of non-degenerate real hypersurfaces S of complex manifolds M for which the groups A(S) of pseudo-conformal transformations of S have either the largest dimension n2 + 2n or the second largest dimension.

One of the most challenging problems in complex differential geometry is the following conjecture of Frankel [3].

(F-n) A compact Kaehler manifold M of dimension n with positive sectional (or more generally, positive holomorphic bisectional) curvature is biholomorphic to the complex projective space Pn(C).

Let k be a field complete with respect to a non-trivial, non-archimedean valuation and let g be a positive integer. Consider the following question : if Γ is a multiplicative subgroup of Gg = (k*)g satisfying certain “Riemann conditions”, can one construct in a natural way an abelian variety defined over k having Gg/Γ as its set of k-rational points? This problem was first considered by Morikawa [3]. J. Tate provided a complete solution for g = 1 (cf. for example [6]). J. McCabe [2] gave a partial solution for g > 1. He showed how to attach to Γ a graded ring R of theta functions such that A = Proj. R is g-dimensional abelian variety over k.

The purpose of this note is to show how the theta-functions attached to certain indefinite quadratic forms of signature (2, 2) can be used to produce a map from certain spaces of cusp forms of Nebentype to Hilbert modular forms. The possibility of making such a construction was suggested by Niwa [4], and the techniques are the same as his and Shintani’s [6]. The construction of Hilbert modular forms from cusp forms of one variable has been discussed by many people, and I will not attempt to give a history of the subject here. However, the map produced by the theta-function is essentially the same as that of Doi and Naganuma [2], and Zagier [7]. In particular, the integral kernel Ω(τ, z1, z2) of Zagier is essentially the ‘holomorphic part’ of the theta-function.

Wir betrachten geschlossene Geodatische c = (c(t), 0 ≤ t ≤ 1) auf Flächen. Der Einfachheit halber nehmen wir an, daß die Fläche kompakt ist, obwohl diese Annahme nur für einen Teil unserer Untersuchungen, insbesondere die Existenzsätze, notwendig ist. Die Geodätische c besitzt einen Index; es ist dies die maximale Anzahl von Parametern, mit der man Variationen von c bilden kann, die sämtlich aus kürzeren Kurven bestehen. Dabei sei hier vorausgesetzt, daß die geschlossene Geodätische c nicht-entartet ist.

We have shown in [1]

Theorem A. Let L be a lattice in a regular quadratic space U over Q; then L has a submodule M satisfying the following conditions 1),2):

• 1) dM ≠ 0, rank M = rank L — 1, and M is a direct summand of L as a module.

• 2) Let L′ be a lattice in some regular quadratic space U′ over Q satisfying dL′ = dL, rank L′ — rank L, tp(L′) ≥ tp(L) for any prime p. If there is an isometry α from M into L′ such that α(M) is a direct summand of L′ as a module, then L′ is isometric to L.

R. Bowen [1] introduced the notion of pseudo-orbit for a homeomorphism f of a metric space X as follows: A (double) sequence {xi}i∈Z of points Xi in X is called a δ-pseudo-orbit of f iff

d(fxi, xi+1) ≤ δ

for every i ∈ Z, where d denotes the metric in X. We say f is stochastically stable if for every ε > 0 there exists δ > 0 such that every δ pseudo-orbit {Xi}i∈Z of f is ε-traced by some x ∈ X, i.e.,

d(fix, xi) ≤ ε

for every i ∈ Z. He proved in [1] that if a compact hyperbolic set Λ for a diffeomorphism f of a compact manifold M has local product structure then the restriction f | Λ of f to Λ is stochastically stable, using stable and unstable manifolds.

Let G be a simply connected Lie group and P a parabolic subgroup without simple factor. A finite dimensional irreducible representation of P defines a homogeneous vector bundle E over the homogeneous space G/P. Ramanan [2] proved that, if the second Betti number b2 of G/P is 1, the inequality in Definition (2.3) holds provided F is locally free. Since the notion of the H-stability was not established at that time, it was inevitable to assume that b2 = 1 and F is locally free. In this paper, pushing Ramanan’s idea through, we prove that E is H-stable for any ample line bundle H. Our proof as well as Ramanan’s depends on the Borel-Weil theorem. If we recall that the Borel-Weil theorem fails in characteristic p > 0, it is interesting to ask whether our theorem remains true in characteristic p > 0.