Determinant is a most useful tool in every branch of mathematics, especially in linear mathematics. What kind of quantities do take a similar universal important role as determinant, in advanced branches of mathematics? In the present articles, showing the usefulness of semi-invariants in the classical invariant theory, we shall give a partial answer of the above question.

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

In this paper we shall establish some generalization of Nevanlinna’s Lemma on Logarithmic Derivative to the case of meromorphic maps from a finite analytic covering space over the m-dimensional complex affine space ℂm to a smooth complex projective variety. Then we shall apply this to “the Second Main Theorem” in Nevanlinna theory in several complex variables.

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.

It is shown that a Siegel-Hecke eigenform of integral weight k and genus 2 is uniquely determined by its Fourier coefficients indexed by nT where T runs over all half-integral positive definite primitive matrices of size 2 and n over all squarefree positive integers. The proof uses analytic arguments involving Koecher-Maaß series and spinor zeta functions.

In this paper we extend the work on affine immersions [N-Pi]-1 to the case of affine immersions between complex manifolds and lay the foundation for the geometry of affine Kähler immersions. The notion of affine Kähler immersion extends that of a holomorphic and isometric immersion between Kähler manifolds and can be contrasted to the notion of holomorphic affine immersion which has been established in the work of Dillen, Vrancken and Verstraelen [D-V-V] and that of Abe [A].

In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.

Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em.

In this paper, we study higher reciprocity law of irreducible polynomials f(x) over Q of degree 3, especially, its close connection with elliptic curves rational over Q and cusp forms of weight 1. These topics were already studied separately in a special example by Chowla-Cowles [1] and Hiramatsu [2]. Here we bring these objects into unity.

1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are:

• (I) identical representation,

• (II) rationally irreducible representation of degree l – 1,

• (III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

In this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Γ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Γ\H has finite volume yet is non-compact. Associated to each cusp of Γ\H, there is a classically studied first-order non-holomorphic Eisenstein series E(s, z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein series E(s, z) admits a meromorphic continuation with a simple pole at s = 1. Classically, Kronecker’s limit formula is the study of the constant term 1 (z) in the Laurent expansion of E(s, z) at s = 1. A number of authors recently have studied what is known as the second-order Eisenstein series E*(s, z), which is formed by twisting the Dirichlet series that defines the series E(s, z) by periods of a given cusp form f. In the work we present here, we study an analogue of Kronecker’s limit formula in the setting of the second-order Eisenstein series E* (s, z), meaning we determine the constant term 2(z) in the Laurent expansion of E*(s, z) at its first pole, which is also at s = 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function 1. We then define two families of convolution Dirichlet series, denoted by and with m ∈ ℕ, which are formed by using the Fourier coefficients of 1 and the weight two cusp form f. We prove that for all m, and admit a meromorphic continuation and are holomorphic at s = 1. Turning our attention to the second-order Kronecker limit function 2, we first express 2 as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp form f, the Fourier coefficients of the first-order Kronecker limit function 1, and special values (1) and (1) of the convolution Dirichlet series. Finally, we prove a bound for the special values (1) and (1) which then implies a bound for the Fourier coefficients of 2. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.

We consider a signal detection problem for the continuous-time stationary diffusion processes. The optimal decision region is given by Neyman-Pearson’s lemma. We establish certain large deviation estimates, and with the help of it we show that the error probability of the second kind of the signal detection tends to zero or one exponentially fast, depending on the fixed exponent of the decay of the error probability of the first kind, as the observation time goes to infinity.

In this paper, we consider a kind of multilinear operators related to oscillatory singular integrals with rough kernels and give a criterion of certain boundedness for this kind of operators.

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

Let S(m), T(n) be positive definite integral matrices and suppose that T is represented by S over each p-adic integer ring Zp. We proved arithmetically in [3] that T is represented by S over Z provided that m ≥ 2n + 3 and the minimum of T is sufficiently large. This guarantees the existence of at least one representation but does not give any asymptotic formula for the number of representations. To get an asymptotic formula we must employ analytic methods.

Let V be a real vector space. An arrangement of hyperplanes in V is a finite family of hyperplanes of V through the origin. We say that is essential if ∩H ∊H = {0}

Soient X un groupe abélien localement compact et dénombrable à l’infini, et ξ sa mesure de Haar. Dans la théorie du potentiel, un noyau de convolution N sur X signifie une mesure de Radon positive dans X. Pour une mesure de Radon réelle μ dans X, N* μ s’appelle le N-potentiel de μ dès que cette convolution a un sens.

The paper will present some basic properties of minimal fine derivatives which seems to be a new concept (or at least a new combination of well-known ones). Why is it worthwhile to study this new concept? Why hasn’t it been done earlier?

In [17], Weil studied the space of representations of certain Fuchsian groups as a generalization of Jacobian variety. The theory of stable vector bundles over a curve developed by Mumford, Seshadri and others are the theory of unitary representations of Fuchsian groups. The moduli space of stable vector bundles over a curve is the space of the irreducible unitary representations of a Fuchsian group. The moduli space is studied in detail. Recently Mumford (unpubished) and Takemoto [12] introduced the notion of H-stable vector bundle over a non-singular projective algebraic surface. In this paper, we study the space of the irreducible unitary representations of the fundamental group of a hyperelliptic surface. Our view point is based on the theory of H-stable vector bundles of Takemoto [12] and [13]. We deal only with hyperelliptic surfaces. Our results should be generalized to the vector bundles over some other surfaces (See §3).

In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and µ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p < ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.