We systematically develop an algebraic technique of free products of operators and their Green’s functions. We apply this framework to obtain, in a simple and uniform fashion, several results on the spectra of graph Laplaceans and random walks.

In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).

Let $S$ be a smooth minimal complex surface of general type with $p_{g}=0$ and $K^{2}=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^{2}=7$ is isomorphic to $\mathbb{Z}_{2}^{2}$ or $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.

Let A be a real analytic set, M be a compact real analytic manifold and f : A × M → R be a real analytic function. Then we have a family of real analytic functions fa, a ∈ A, on M defined by fa(X) = f(a, x).

Automorphic induction for GLn is a case of endoscopic transfer, and its character identity was established by Henniart and Herb, up to a constant of proportionality. We determine this constant in terms of the Kottwitz-Shelstad normalization of transfer factors, which involves certain ε-factors.

We show that perfectoidization can be (almost) calculated by using p-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidizations and uniform completions, as well as the p-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation “p-root closure” used by Roberts in mixed characteristic commutative algebra and a more recent concept of “perfectoidization” introduced by Bhatt and Scholze in their theory of prismatic cohomology.

We study the Fourier coefficients of cusp forms of half integral weight and generalize the Kohnen-Zagier formula to the case of ‘4 × general odd’ level by using results of Ueda. As an application, we obtain a generalization of the result of Luo-Ramakrishnan [11] to the case of arbitrary odd level.

A density P on the punctured unit disk Ω:0 < |z| <1 is a 2-form P(z)dxdy whose coefficient P(z) is a real valued nonnegative locally Hölder continuous function on the closed punctured unit disk Ω:0< |z| <≦1. Here we consider Ω as an end of the punctured sphere 0 < |z| ≦ + + so that the point z = 0 is viewed as the ideal boundary δΣ of Σ and the unit circle |z| = 1 as the relative boundary δΣ of Σ. We denote by D = D(Σ) the family of densities on Σ.

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a nonarchimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin–Selberg $L$-factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$ is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$. As an application we prove that the $L$-factor of a pair of banal $\ell$-modular cuspidal representations is the reduction modulo $\ell$ of the $L$-factor of any pair of $\ell$-adic lifts.

A holomorphic n × n-matric automorphic factor with respect to a lattice L in Cg means a system of holomorphic n × n-matrices {μα(z) | α ∈ L} such that

The purpose of this paper is to discuss some ergodic properties of the stepping stone model proposed by Kimura, M. [4] and developed by Weiss, G.H. and Kimura, M. [12]. Our model to be discussed in this paper involves selection in addition to mutation and migration which are dealt with in [4], [12]. Because of the additional factor selection, the stochastic process describing our model becomes complicated and presents particularly interesting profound structure of the random phenomena in question.

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$ , where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$ .

Recently Mr. Seizô Itô kindly called the author’s attention to the fact that the derivation of (4.15) in the above referred paper is insufficient since the paper does not contain the proof of the compactness condition:

(1)

uniformaly in a and b

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

Let Q be the rational number field, K/Q be a finite Galois extension with the Galois group G, and let CK be the ideal class group of K in the wider sense. We consider CK as a G-module. Denote by I the augmentation ideal of the group ring of G over the ring of rational integers. Then CK/I(CK) is called the central ideal class group of K, which is the maximal factor group of CK on which G acts trivially. A. Fröhlich [3, 41 rationally determined the central ideal class group of a complete Abelian field over Q whose degree is some power of a prime. The proof is based on Theorems 3 and 4 of Fröhlich [2]. D. Garbanati [6] recently gave an algorithm which will produce the l-invariants of the central ideal class group of an Abelian extension over Q for each prime l dividing its order.

In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”

Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system.

In the previous paper [6], we have considered flows on the measure space (Ω, ℬ, P), which was the projective limit of a certain subspace (Ωn, ℬn, Pn) of the measure space (Sn, ℬ(Sn), Pn), where Sn is the (n − 1) · sphere with radius and Pn is the uniform probability distribution over Sn.

Several years ago, K. Oikawa, my colleague, investigated the properties of Schottky coverings of closed Riemann surfaces, leaving an interesting problem as open [3]:

Does exist a Schottky covering between the basic surface and a given planar covering?

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.