This paper examines the moduli spaces of log Hodge structures introduced by Kato and Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. This paper treats the following two cases: (A) where the period domain is Hermitian symmetric, and (B) where the Hodge structures are of the mirror quintic type. Especially it addresses a property of the torsor.

The fundamental germ is a generalization of π1, first defined for laminations which arise through group actions [4]. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the PSL(2,ℤ) lamination are calculated, laminations for which the definition in [4] was not available. The mother germ is used to give a new proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type.

In their paper [11], Kubota-Leopoldt constructed the p-adic L-function Lp(s; χ) for each Dirichlet character χ. This function Lp(s; χ) is a mero-morphic function on the open dies {|s − 1| < |<q-1 p1/(p-1)|} with a possible simple pole at s = 1.

L’étude sur les directions de Julia des fonctions méromorphes dans le plan fini est fait présque complètement par Ostrowski [5] et après Rauch [6] etc. l’ont développé.

Cet article fait suite à [3c]; dans [3c] étaient étudiées des généralisations éventuelles du principe de minimum fin de Fuglede (voir [2] théo. 9.1. et [3a] théo. 6.) et le cardre était celui des processus de Markov en dualité au sens de Kunita-Watanabe (voir [4b]). Le présent article a le même but, si dans [3c] on a supposé que la limite inférieure fine de la fonction finement hyperharmonique donnée doit être non négative en tout point de la fontière fine de l’ensemble de définition, on se permet dans cet article de supposer que cette lim. inf. fine pourra avoir des valeurs négatives (même égales à — ∞) sur un ensemble polaire de la fontière fine.

Dans ce mémoire, on fera pour les fonctions algébroïdes des études analogues à ceux des fonctions méromorphes dans |z|<∞ faits dans Doob [3], Toda [7].

Soit g(z) une fonction méromorphe dans |Z|<∞ ayant un point singulier essentiel isolé à l’infini. J.L. Doob a obtenu le théorème suivant en utilisant la topologie fine pour g(z).

Soit $\unicode[STIX]{x1D70B}$ un module de plus haut poids unitaire du groupe $G=\mathbf{Sp}(2n,\mathbb{R})$. On s’intéresse aux paquets d’Arthur contenant $\unicode[STIX]{x1D70B}$. Lorsque le plus haut poids est scalaire, on détermine les paramètres de ces paquets, on établit la propriété de multiplicité $1$ de $\unicode[STIX]{x1D70B}$ dans le paquet, et l’on calcule le caractère $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D70B}}$ (du groupe des composantes connexes du centralisateur du paramètre dans le groupe dual) associé à $\unicode[STIX]{x1D70B}$ et qui joue un grand rôle dans la théorie d’Arthur. On fait de même pour certains modules de plus haut poids unitaires unipotents $\unicode[STIX]{x1D70E}_{n,k}$, ou bien lorsque le caractère infinitésimal est régulier.

The following is a well-known result due to A.P. Calderon [2], asserting the existence of non-tangential limits of multiply harmonic functions.

Let k be a field, and X = [xij] be an n × (n + m) matrix whose elements are algebraically independent over k.

We shall study the canonical module of the graded ring R, which is a quotient ring of the polynomial ring A = k[X] by the ideal αn(X) generated by all the n × n minors of X.

In this paper we first prove a simple relation between sums of a certain type and quadratic residue symbols. Then we apply this to Dedekind sums introduced by Sczech [5]. In particular one of his conjectures in [6] will be proved.

Für jede Riemannsche Fläche R bezeichnen wir mit M(R) die Algebra von Royden auf R und mit R* (bzw. ΔR)- die Kompaktifizierung (bzw. den idealen Rand) von Royden von R ([5]). Für jede Funktion ξ ∊ M(R) sei Jξ das kompakte Segment und X(R) der kompakte Raum . Laut der Definition von R* ist jede Funktion ξ∊M(R) auf R* (in eindeutiger Weise) stetig fortsetzbar. Bezeichnet man ebenfalls mit ξ die so fortgesetzte Funktion und mit φR die Abbildung von R*→X(R), so ist φR ein Homeomorphismus von R* auf φR(R*).

The Riemann’s theta functions associated with a closed Riemann surface are absolutely convergent. In the present paper, we shall show an example of an hyperelliptic Riemann surface of infinite genus such that the Riemann’s theta functions associated with are absolutely convergent.

Let p be a prime. Let k be a -adic number field and be the ring of all integers of k. Let K/k be a cyclic totally ramified extension of degree pn with Galois group G. Clealy the ring of all integers of K is an [G]-module, and the purpose of this paper is to give a necessary and sufficient condition for the [G]-module to be indecomposable.

The main objective of this paper is to prove the following theorem, which generalizes some results of [1], [2], [6]. Our theorem is also suggested by the work of Novikov [5].

The main purpose of this paper is to investigate the parallelism of vectors in homogeneous spaces. The definition of a vector and the condition for spaces under which a covariant differential of a vector is also a vector were given by E. Cartan [4] in a very intuitive way. Here I formulate this in a stricter way by his method of moving frame. Even if a homogeneous space has the property that the covariant differential of a vector is of the same kind, another definition of covariant differential may also have the required property. I will give a necessary and sufficient condition under which the definition of covariant differential is unique. Once the covariant differential has been defined it is easy to introduce a parallelism of vectors in the space. But the parallelism depends in general on the path along which we translate a vector. The condition for the spaces with an absolute parallelism can be obtained.

Let f: X → Y be a finite type morphism of locally noetherian schemes. It is well known ([3, IV, 7.8.6]) that the excellent property ascends from Y to X. On the other side there are counter-examples where X is excellent and Y is not. First of all it is easy to show that the condition on chains of prime ideals does not descend (see [3, IV, 7.8.4]), even by finite morphisms. Secondly in [2] it is produced an example where X is excellent while Y is not a G-scheme (i.e. it has not the good properties of formal fibers). However in [2] it is also proved that the property concerning the openness of regular loci (the so called “J-2”) descends by finite type surjective morphisms. Therefore we are led to the following question: When does the G-scheme property descend? I.e. what conditions do we need on f? A reasonable condition is conjectured (in [2]) as the following: f is proper surjective. The aim of the present paper is precisely to give an answer to such a question. What we really prove is the following. If X is a G-scheme and J-2 (quasi-excellent), then the same is true for Y, provided that f is proper surjective and moreover all the residue fields of Y have characteristic 0. We remark that the result is strongly based on Hironaka’s desingularization for quasi-excellent schemes defined over a field of characteristic 0.

Given an arbitrary Riemannian n-space V let σ be a harmonic field in the complement V-V0 of a regular region V0. The problem of constructing in v a harmonic field ρ with the property was given a complete solution in [2]. The corresponding problem for harmonic forms σ, ρ remains open iri the general case. In the special case of locally flat spaces the construction can be carried out by replacing by the point norm [3].

For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the field

K = Q(ζm + ζm−1)

respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

The general problem of producing concrete representations for continuous linear functionals on normed linear spaces, ie., of identifying conjugate spaces, has of course attracted the attention of many mathematicians during the last five decades and has been solved in many cases [1, pp. 59-72]. Likewise, the problem of extending a linear functional defined on a linear subspace of a normed linear space may be regarded as solved by the Hahn-Banach theorem [1, p. 28], although problems involving “natural” extensions, like that yielding the Lebesgue integral from the Riemann integral, remain. In the present paper, we shall consider two “natural’ methods of extending a certain linear functional and show that they are in fact identical. As a by-product, we obtain a concrete representation both for the original functional and for its “natural” extension. In subsequent communications, the writers will consider topologies in certain families of linear functionals, canonical resolutions of linear functionals, and other extension problems.

Let D denote the unit disk | z | < 1, and C the unit circle | z | = 1. Corresponding to any function f meromorphic in D we denote by f* the spherical derivative