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.

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).

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 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.

Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant K-group of Steinberg’s triple variety. The K-group is equipped with a filtration indexed by closed G-stable subvarieties of the nilpotent variety, where G is the corresponding reductive algebraic group over ℂ. In this paper we will show in the case of type A that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

We show equi-distribution properties of values for the third and the fifth Painlevé transcendents in a sectorial domain. For our purpose we define a characteristic function of sectorial domain type by employing value distribution theory in a half plane. Some special cases admit analogues of Borel exceptional values. Similar results are obtained for modified versions of these Painlevé transcendents, which are of infinite growth order.

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:

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.

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 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.

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.

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.

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.

The topic of this note is the Riesz decomposition of excessive functions for a “nice” strong Markov process X. I.e. an excessive function is decomposed into a sum of a potential of a measure and a “harmonic” function. Originally such decompositions were studied by G.A. Hunt [8]. In [1] a Riesz decomposition is given assuming that the state space E is locally compact with a countable base and X is a transient standard process in strong duality with another standard process having a strong Feller resolvent. Recently R.K. Getoor and J. Glover extended the theory to the case of transient Borei right processes in weak duality [6].

For a smooth proper variety over a p-adic field, its Brauer group and abelian fundamental group are related to higher Chow groups by the Brauer–Manin pairing and class field theory. We generalize this relation to smooth (possibly nonproper) varieties, using motivic homology and a variant of Wiesend’s ideal class group. Several examples are discussed.

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.

Classically, trace was defined as the sum of the diagonal entries of a square matrix with entries in a field. This notion played an important role in classical mathematics, e.g. in the theory of algebras over a field of characteristic zero, and in the theory of group characters (as in [1]). A generalization to endomorphisms of a finitely generated projective module over any ring R with unit is well-known. For such a module P the canonical homomorphism Ψ: P* ⊗ P→ EndR(P) is an isomorphism. Then the composite ε˚Ψ-1: EndR(P) → P* ↗ P→ R, where e denotes “evaluation”, is a homomorphism which coincides with the classical trace whenever P is free. This version of trace has been used by Hattori [3] and others to study projective modules. However, this approach to trace is limited to the finitely generated projective modules, since it can be shown that Ψ is an isomorphism if and only if P is finitely generated and projective.

Let $K=\mathbb{F}_{q}(T)$ and $A=\mathbb{F}_{q}[T]$. Suppose that $\unicode[STIX]{x1D719}$ is a Drinfeld $A$-module of rank $2$ over $K$ which does not have complex multiplication. We obtain an explicit upper bound (dependent on $\unicode[STIX]{x1D719}$) on the degree of primes ${\wp}$ of $K$ such that the image of the Galois representation on the ${\wp}$-torsion points of $\unicode[STIX]{x1D719}$ is not surjective, in the case of $q$ odd. Our results are a Drinfeld module analogue of Serre’s explicit large image results for the Galois representations on $p$-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld $A$-modules of rank $2$ over $K$ is also proven.