Among the most remarkable results in the theory of harmonic functions on Riemann surfaces is the strictness of the inclusion relations OG < OHB < OHD, established by Ahlfors [1,2], Royden [2,4], and Tôki [8] two decades ago. Subsequently the strictness of the relations OG < OHP < OHB was shown and a somewhat simpler proof of OHB < OHD given by Sario [5] and Tôki [9]. Here OG is the class of parabolic surfaces, and OHP, OHB, OHD stand for the classes of surfaces which do not carry nonconstant harmonic functions which are positive, bounded, or Dirichlet finite, respectively.

We construct a stable formal model of a Lubin–Tate curve with level three, and study the action of a Weil group and a division algebra on its stable reduction. Further, we study a structure of cohomology of the Lubin–Tate curve. Our study is purely local and includes the case where the characteristic of the residue field of a local field is two.

We prove Iitaka $C_{n,m}$ conjecture for $3$-folds over the algebraic closure of finite fields. Along the way we prove some results on the birational geometry of log surfaces over nonclosed fields and apply these to existence of relative good minimal models of $3$-folds.

Let Λ be a ring and G a finite group of ring automorphisms of Λ. The totality of elements of Λ which are left invariant by G is a subring of Λ. We call it the G-fixed subring of Λ. Let be the crossed product of Λ and G with trivial factor set, i.e. {u0} is a Λ-free basis of Δ and , and let Γ be a subring of the G-fixed subring of Λ which has the same identity as Λ.

Recently M. G. Crandall and P. L. Lions developed the viscosity theory on nonlinear equations in infinite dimensions and optimal control in Hilbert spaces, in two series of papers, [1], [4].

Expansions of generalized functions have been investigated by many authors. Korevaar [11], Widlund [20], Giertz [8], Walter [19] developed procedures for expanding generalized functions of Korevaar [12], Temple [17], and Lighthill [13], Expansions of certain Schwartz distributions [15] into series of orthonormal functions were given by Zemanian [23] (see also Zemanian [24]) and thereby he extended a number of integral transforms to distributions. The method involved in his work is very much related to the Hilbert space technique and is of somewhat different character from those used in most of the works on integral transforms such as [24, chapters 1-8]. Other works that discuss orthogonal series expansions involving generalized functions are by Bouix [1, chapter 7], Braga and Schönberg [2], Gelfand and Shilov [7, vol. 3, chapter 4] and Warmbrod [21].

The purpose of this paper is to prove functional equations of L-functions associated with the vector space of binary quadratic forms and determine their poles and residues. For a commutative ring K, let V(K) be the set of all symmetric matrices of degree 2 with coefficients in K. In V(C), we consider the inner product

where for .

M. Riesz [8] introduced the notion of α-superharmonic functions in n(≥1)-dimensional Euclidean space Rn in connection with the potential of order α. In this paper, we shall first define the α-superharmonic and α-harmonic functions in a domain D. In case α = 2, they coincide with ones in the usual sense.

For a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).

In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

In this paper, we shall supply proofs to the results announced in [2], pp. 74-75: we shall prove the Siegel formula for the Pfaffian of degree n over an algebraic number field and also determine the zeta function of the Pfaffian. In the appendix, we shall briefly discuss the non-split case where the Pfaffian is replaced by the norm form of the simple Jordan algebra of quaternionic hermitian matrices of degree n.

This paper is a continuation of two previous papers of the author. In the first [4] we discussed a Thompson series associated with the group M24 in which each of the modular forms ηg(τ) attached to elements g ∈ M24 are primitive cusp-forms.

Let dz(n) be a multiplicative function defined by

where s = σ + it, z is a. complex number, and ζ(s) is the Riemann zeta function. Here ζz(s) = exp(z log ζ(s)) and let log ζ(s) take real values for real s > 1. We note that if z is a natural number dz(n) coincides with the divisor function appearing in the Dirichlet-Piltz divisor problem, and d-1(n) with the Möbious function.

Let k be an integral domain, let F = (F1X, Y),…, Fn(X, Y)), X = (X1,…, Xn), Y = (Y1,…, Yn), be an n-dimensional formal group over k, and let L(F) be the Lie algebra of all F-invariant k-derivations of the ring of formal power series k[X] (cf. § 2). If A is a (commutative) k-algebra and Derk (A) denotes the Lie algebra of all k-derivations d: A → A, then by an action of L(F) on A we mean a morphism of Lie algebras φ: L(F) → Derk (A) such that φ(dp) = φ(d)p, provided char (k) = p > 0. An action of the formal group F on A is a morphism of k-algebras D: A-→A[X] such that D(a)≡a mod (X) for a ∊ A, and FA ∘ D = DY ∘ D, where FA: A[X] → A[X, Y], DF: A[X] → A[X, Y] are morphisms of k-algebras given by FA(g(X)) = g(F), DY(Σa aaXa) = Σa D(aa)Y; for a motivation of this notion, see [15]. Let D: A → A[X] be such an action.

In the theory of meromorphic functions, it is important to investigate the properties of covering surfaces generated by their inverse functions. For this purpose, the study of properties of a non-compact region of a Riemann surface is useful.

This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$. We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

Let fλ(z) = λez. In this short note, we consider those maps fλ with λ close to 1. We show that the probability that fλ is hyperbolic approaches 1 as λ → 1.

We consider the problem of whether the norm one torus defined by a finite separable field extension K/k is stably (or retract) rational over k. This has already been solved for the case where K/k is a Galois extension. In this paper, we solve the problem for the case where K/k is a non-Galois extension such that the Galois group of the Galois closure of K/k is nilpotent or metacyclic.

In [7], O. Lehto and K. I. Virtanen introduced the concept of normal meromorphic functions in connection with the study of boundary behaviour of meromorphic functions of one complex variable.

In this paper, we generalize the theory of normal meromorphic functions to the case of holomorphic mappings into higher dimensional complex spaces in connection with the theory of hyperbolic manifolds and Nevanlinna theory.

The main concern of this paper is the generalizations of the big Picard theorem and Lindelöf’s theorem which appear in the classical function theory.