1. Let F be an algebraic number field of finite degree, and let K/F be a normal extension of degree n. Denote by OK the ring of all integers in K. In we proved the following:

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$, and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

The purpose of this paper is to study threefolds X, with negative Kodaira dimension k(X) and positive irregularity q(X), defined over the complex field C.

We construct a measure over the string bundle associated to the loop space of a Riemannian manifold. We deduce a representation of a finite energy Kac-Moody group analoguous to the energy representation.

In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r > 1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

Let X be a projective Gorenstein variety, Y ⊂ X a proper closed subscheme such that X is smooth at all points of Y, so that the formal completion of X along Y is regular.

In the author’s previous paper [3], we used Theorem 1 of the present paper to assure the existence of a signed branching Markov process with age satisfying given conditions in [3], The purpose of this paper is to give a proof of Theorem 1.

Let P be a pseudodifferential operator of the form

where s, b ≥ 0 are even integers and is odd function with f′(t) > 0 (t ≠ 0). Here . We shall call P an operator of Egorov type because P with f(t) = tk, (k odd) is an important model of subelliptic operators studied by Egorov [1] and Hörmander [3], [4, Chapter 27]. Roughly speaking, any subelliptic operator can be reduced to this operator or Mizohata one after several steps of microlocalization arguments. In this paper we shall study the hypoellipticity of P and the local solvability of adjoint operator P* in the case where f(t) vanishes infinitely at the origin and moreover consider the case where ts and are replaced by functions with zero of infinite order.

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

1. Let Ω be a bounded domain of Rn with boundary ∂Ω and let E be a compact subset of Ω. We assume that both ∂Ω and E have positive capacity. The norm and the inner product in L2(Ω) are simply denoted by ‖ ‖ and (,) respectively.

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.

Heisenberg group H2g+1(R) of dimension 2g + 1 is a real nilpotent group defined on R × Rg × Rg by the law of composition

which is isomorphic to the unipotent matrix group

Let M be a compact non-singular real affine algebraic variety and let A, B be open disjoint semialgebraic subsets of M. Define (where —4 denotes the Zariski closure).

A pair (X, D) of a projective variety X and a very ample divisor D on X is called stable (resp. semi-stable, resp. unstable) if the Chow point corresponding to the embedding is SL(N + 1)-stable (resp. semi-stable, resp. unstable). The criterion for stability is one of the most important steps in proving the existence of moduli spaces.

Let k be an algebraic number field with the ring of integers ok = o and let G be a cyclic group of order p, an odd prime.

The evaluation of the integral

(1.1)

represents one of the fundamental problems of the theory of the Riemann zeta-function (see [4] for a comprehensive account). In view of the functional equation

it is clear that one has to distinguish between the following three principal cases:

• a) σ = 1/2 (“the critical line”),

• b) 1/2 < σ < 1 (“the critical strip”),

• c) σ = 1.

In this paper it is proved that every locally strictly localizable Radon measure of type (ℋ) is strictly localizable, from where it follows immediately the existence of lifting for these measures.

1. Nous avons un théorème, ce qu’on appelle le théorème de Cartan-Dieudonné, sur les générateurs du groupe orthogonal: Toute transformation orthogonale à n variables sur un corps de caractéristique ≠2 est un produit de n symetries au plus, et sur un corps de caractéristique 2 elle est un produit au plus de n transvections orthogonales, sauf un seul cas, [1] ou [3]. C’est une généralisation d’un résultat obtenu par E. Cartan, relatif au corps des nombres réels ou au corps des nombres complexes [4].

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.