Let (G, V) be an irreducible prehomogeneous vector space defined over a number field k, P ∈ k[V] a relative invariant polynomial, and χ a rational character of G such that . For , let Gx be the stabilizer of x, and the connected component of 1 of Gx . We define L0 to be the set of such that does not have a non-trivial rational character. Then we define the zeta function for (G, Y) by the following integral

where Φ is a Schwartz-Bruhat function, s is a complex variable, and dg” is an invariant measure.

Soit C une courbe canonique de genre g ≥ 4. Le théorème de Enriques-Babbage [ACGH] affirme que l’idéal de C est engendré par (g − 2) (g − 3)/2 hypersurfaces quadriques sauf si C est trigonale ou isomorphe à une quintique plane. Si C est trigonale, elle est tracée sur une surface réglée rationnelle normale dont les génératrices découpent la série trigonale. Si C est isomorphe à une quintique plane lisse, elle est tracée sur une surface de Veronése.

Affine Kac-Moody algebras represent a well-trodden and well-understood littoral beyond which stretches the vast, chaotic, and poorly-understood ocean of indefinite Kac-Moody algebras. The simplest indefinite Kac-Moody algebras are the rank 2 Kac-Moody algebras (a) (a ≥ 3) with symmetric Cartan matrix , which form part of the class known as hyperbolic Kac-Moody algebras. In this paper, we probe deeply into the structure of those algebras (a), the e. coli of indefinite Kac-Moody algebras. Using Berman-Moody’s formula ([BM]), we derive a purely combinatorial closed form formula for the root multiplicities of the algebra (a), and illustrate some of the rich relationships that exist among root multiplicities, both within a single algebra and between different algebras in the class. We also give an explicit description of the root system of the algebra (a). As a by-product, we obtain a simple algorithm to find the integral points on certain hyperbolas.

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.

Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

Let p be an odd prime and d be a positive integer prime to p such that d ≢ 2 mod 4. For technical reasons, we also assume that . For each integer n ≥ 1, we choose a primitive nth root ζn of 1 so that whenever n | m. Let be its cyclotomic Zp-extension, where is the nth layer of this extension. For n ≤ 1, we denote the Galois group Ga\(Kn/K0 ) by Gn , the unit group of the ring of integers of Kn by En , and the group of cyclotomic units of Kn by Cn . For the definition and basic properties of cyclotomic units such as the index theorem, we refer [6] and [7]. In this paper we examine the injectivity of the homomorphism between the first cohomology groups induced by the inclusion Cn → En .

Let A be a Weil algebra. The fibre bundle TAM of A-velocities over a manifold M was described by A. Morimoto [15] as another description of the bundle of near A-points by Weil [17]. In [4] for any tensor field τ of type (0,2) on M and any functional λ ∈ A* we have defined the so called λ-lift of τ to TAM. We recall this construction in Example 1.3. The λ-lift of τ is a naturally induced tensor field of type (0,2) on TAM.

S. Bochner proved the following theorem in [B].

In this paper we consider the following problem: Given a smooth function K on the n-dimensional unit sphere Sn(n ≥ 3) with its canonical metric g0 , is it possible to find a pointwise conformal metric which has K as its scalar curvature? This problem was presented by J. L. Kazdan and F. W. Warner. The associated problem for Gaussian curvature in dimension 2 had been presented by L. Nirenberg several years before.

Let Ω be a smoothly bounded pseudoconvex domain in Cn and let A(Ω) denote the functions holomorphic on Ω and continuous on . A point p ∈ bΩ is a peak point if there is a function f ∈ A(Ω) such that f (p) = 1, and | f(z) | < 1 for z ∉ Ω − {p}.