Volume 113, Issue 2
May 2003, pages 91-212
pp 91-98
Some Functional Equations Originating from Number Theory
We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.
pp 99-137
This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. Let 𝐾 be a local field with finite residue class field $k_K$. We first define (cf. Definition 2.4) the conductor $\mathfrak{f}(E/K)$ of an arbitrary finite Galois extension 𝐸/𝐾 in the sense of non-abelian local class field theory as
$$\mathfrak{f}(E/K)=\mathfrak{p}_K^{[[n_G]]+1},$$
where $n_G$ is the break in the upper ramification filtration of $G = \mathrm{Gal}(E/K)$ defined by $G^{n_G} ≠ G^{n_G+𝛿} = 1, \forall𝛿 \in \mathbb{R}_{\gneq 0}$. Next, we study the basic properties of the ideal $\mathfrak{f}(E/K)$ in $O_K$ in case $E/K$ is a metabelian extension utilizing Koch–de Shalit metabelian local class field theory (cf. [8]).
After reviewing the Artin character $a_G:G → \mathbb{C}$ of $G:=\mathrm{Gal}(E/K)$ and Artin representations $A_G:G → GL(V)$ corresponding to $a_G:G → \mathbb{C}$, we prove that (Proposition 3.2 and Corollary 3.5)
$$\mathfrak{f}\mathrm{Artin}(𝜒_ρ)=\mathfrak{p}_K^{\dim_\mathbb{C}(V)[n_{G/\ker(ρ)}+1]},$$
where 𝜒_{ρ} : $G → \mathbb{C}$ is the character associated to an irreducible representation $ρ : G → GL(V)$ of $G (\text{over} \mathbb{C})$. The first main result (Theorem 1.2) of the paper states that, if in particular, $ρ : G → GL(V)$ is an irreducible representation of $G (\text{over} \mathbb{C})$ with metabelian image, then
$$\mathfrak{f}\mathrm{Artin}(𝜒_ρ)=\mathfrak{p}_K^{[E^{\ker(ρ)^\bullet}:K](n_{G/\ker(ρ)}+1)},$$
where $\mathrm{Gal}(E^{\ker(ρ)}/E^{\ker(ρ)^\bullet})$ is any maximal abelian normal subgroup of $\mathrm{Gal}(E^{\ker(ρ)}/K)$ containing $\mathrm{Gal}(E^{\ker(ρ)}/K)'$, and the break $n_{G/\ker(ρ)}$ in the upper ramification filtration of $G/\ker(ρ)$ can be computed and located by metabelian local class field theory. The proof utilizes Basmaji's theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]).
We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a `natural' $A_G$ of 𝐺 over $\mathbb{C}$ (Problem 1.3). More precisely, we prove in Theorem 1.4 that if 𝐸/𝐾 is a metabelian extension with Galois group 𝐺, then
\begin{align*}A_G\simeq & \sum\limits_N[(E^N)^\bullet :K](n_{G/N}+1)\\ & ×\sum\limits_{[𝜔]\sim\in𝜈_N}\mathrm{Ind}_{𝜋_N^{-1}((G/N)^\bullet)}^G\left(𝜔\circ𝜋_N|_{𝜋_N^{-1}((G/N)^\bullet)}\right),\end{align*}
where 𝑁 runs over all normal subgroups of 𝐺, and for such an $N, 𝜈_N$ denotes the collection of all ∼-equivalence classes $[𝜔]_\sim$, where `∼' denotes the equivalence relation on the set of all representations $𝜔 :(G/N)^\bullet→\mathbb{C}^×$ satisfying the conditions
$$\mathrm{Inert}(𝜔)=\{𝛿\in G/N :𝜔_𝛿=𝜔\}=(G/N)^\bullet$$
and
$$\bigcap\limits_𝛿 \ker(𝜔_𝛿)=\langle 1_{G/N}\rangle,$$
where 𝛿 runs over $\mathcal{R}((G/N)^\bullet\backslash(G/N))$, a fixed given complete system of representatives of $(G/N)^\bullet\backslash(G/N)$, by declaring that $𝜔_1 \sim 𝜔_2$ if and only if 𝜔_{1} = 𝜔_{2,𝛿} for some $𝛿 \in \mathcal{R}((G/N)^\bullet\backslash(G/N))$.
Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.
pp 139-152
The Jacobian of a Nonorientable Klein Surface
Pablo Arés-Gastesi Indranil Biswas
Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface 𝑌 is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms on 𝑌 quotiented by the torsion-free part of the first integral homology of 𝑌. Denote by 𝑋 the double cover of 𝑌 given by orientation. The Jacobian of 𝑌 is identified with the space of all degree zero holomorphic line bundles 𝐿 over 𝑋 with the property that 𝐿 is isomorphic to $𝜎^*\overline{L}$, where 𝜎 is the involution of 𝑋.
pp 153-163
Reduction Theory for a Rational Function Field
Let 𝐺 be a split reductive group over a finite field $F_q$. Let $F = F_q(t)$ and let 𝐴 denote the adèles of 𝐹. We show that every double coset in $G(F)\backslash G(A)/K$ has a representative in a maximal split torus of 𝐺. Here 𝐾 is the set of integral adèlic points of 𝐺. When 𝐺 ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.
pp 165-169
A Note on Absolute Summability Factors
In this paper, by using an almost increasing and 𝛿-quasi-monotone sequence, a general theorem on $\varphi - |C, 𝛼|_k$ summability factors, which generalizes a result of Bor [3] on $\varphi-|C, 1|_k$ summability factors, has been proved under weaker and more general conditions.
pp 171-178
Wavelet Subspaces Invariant Under Groups of Translation Operators
Biswaranjan Behera Shobha Madan
We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.
pp 179-182
Let 𝑓 be a continuous function on the unit circle 𝛤, whose Fourier series is 𝜔-absolutely convergent for some weight 𝜔 on the set of integers $\mathcal{Z}$. If 𝑓 is nowhere vanishing on 𝛤, then there exists a weight 𝜈 on $\mathcal{Z}$ such that 1/𝑓 had 𝜈-absolutely convergent Fourier series. This includes Wiener's classical theorem. As a corollary, it follows that if 𝜑 is holomorphic on a neighbourhood of the range of 𝑓, then there exists a weight 𝜒 on $\mathcal{Z}$ such that $\varphi\circ f$ has 𝜒-absolutely convergent Fourier series. This is a weighted analogue of Lévy's generalization of Wiener's theorem. In the theorems, 𝜈 and 𝜒 are non-constant if and only if 𝜔 is non-constant. In general, the results fail if 𝜈 or 𝜒 is required to be the same weight 𝜔.
pp 183-187
Fixed Point of Multivalued Mapping in Uniform Spaces
In this paper we prove some new fixed point theorems for multivalued mappings on orbitally complete uniform spaces.
pp 189-193
The Weierstrass–Enneper Representation using Hodographic Coordinates on a Minimal Surface
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass–Enneper representation, using a system of hodographic coordinates. This is done by using the method of solving the Born–Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass–Enneper representation. From this we derive the hodographic coordinate $ρ \in D \subset \mathbb{C}$ and 𝜎 its complex conjugate which enables us to write the Weierstrass–Enneper representation in a new way.
pp 195-205
Positive Solutions of Singular Boundary Value Problem of Negative Exponent Emden–Fowler Equation
This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden–Fowler equation. A necessary and sufficient condition for the existence of 𝐶[0, 1] positive solutions as well as 𝐶^{1}[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.
pp 207-212
Bounds on the Phase Velocity in the Linear Instability of Viscous Shear Flow Problem in the 𝛽-Plane
Results obtained by Joseph (J. Fluid Mech. 33 (1968) 617) for the viscous parallel shear flow problem are extended to the problem of viscous parallel, shear flow problem in the beta plane and a sufficient condition for stability has also been derived.
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