Let Ω be a bounded domain in ndimensional Euclidian space Rn(n ≧ 2)with C∞boundary Γ of dimension n— 1 and let there be given two real-valued C∞–functions α, β on Γ such that α≧ 0, β ≧ 0and α + β = lthroughout Γ.Then we consider the non-elliptic boundary value problem with λ ≧ 0 (which is always assumed, and in particular when λ =0, we further assume β ≢ 0,throughout this paper) :

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To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef–Loeser formulas for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

In this paper we use a general technique, due to Larfeldt-Lech [9], to show that there exist local noetherian rings which are “bad” with respect to some properties which are obviously verified in the algebro-geometric case. In particular we show that there exist local Gorenstein rings which are not homomorphic images of regular (local) rings and that there exist local rings which do have canonical module but do not have canonical algebra.

Let X be a locally compact, σ-compact and non-compact abelian group. Throughout this paper, we shall denote by ξ a fixed Haar measure on X and by δ the Alexandroff point of X.

Let E be a totally disconnected compact set in the z-plane and let Ω be its complement with respect to the extended 2-plane. Then Ω is a domain and we can consider a single-valued meromorphic function f(z) in Ω which has a transcendental singularity at each point ζ ∊ E. Suppose that E is a null-set of the class W in the sense of Kametani [4] (= the class NB in the sense of Ahlfors and Beurling [1]).

Let Rn be the Euclidean space of dimension n ≧ 1 with the standard inner product and the norm be the unit sphere {x ∈ Rn; |x| = 1} and dωn-1 be the volume element of Sn-1 such that Sn-1 gets the volume 1. Let Ω be an open set of Rn containing Sn-1 and let f: Ω → Rm be a smooth map.

Wir betrachten geschlossene Geodatische c = (c(t), 0 ≤ t ≤ 1) auf Flächen. Der Einfachheit halber nehmen wir an, daß die Fläche kompakt ist, obwohl diese Annahme nur für einen Teil unserer Untersuchungen, insbesondere die Existenzsätze, notwendig ist. Die Geodätische c besitzt einen Index; es ist dies die maximale Anzahl von Parametern, mit der man Variationen von c bilden kann, die sämtlich aus kürzeren Kurven bestehen. Dabei sei hier vorausgesetzt, daß die geschlossene Geodätische c nicht-entartet ist.

In the theory of modular forms there is an interesting problem whether every modular form can be expressed as a linear comination of theta functions. For this Eichler proved in [1] that for a sufficiently large prime q all modular forms of degree — 2m(m = 1,2, · · ·) for Γ0(q) can be represented by linear combinations of theta functios of degree — 2m with level 1 and q. We prove this theorem for q = 2, 3, 5 and 11 by using a theorem of Siegel for q = 2, 3, 5 and a general result of Eichler for q = 11. The former method is shown in Schoeneberg [2].

This paper is concerned with the cohomology at “infinity” (in the sense of Harder [4], [5]) of a congruence subgroup of the symplectic group G = Sp(2l, R). G is the subgroup of GL(2l, R) consisting of matrices g satisfying tgJg = J where

We prove that the maximal number of conics in a smooth sextic $K3$ -surface $X\subset \mathbb {P}^4$ is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible.

The examples on p. 44 of metabelian Lie algebras L1 and L2 are isomorphic over certain base fields contrary to the claim in the paper. More specifically, the error lies in the incorrect statement that the rank of ad x for any x in L1 is either zero or is at least 3. This statement does hold over Q, but if τ is a scalar satisfying τ2 − τ − 1 = 0 then ad (x1 + τx2 + τ2x3 + x4 + τ2x5 has rank 2.

Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell }$, with $\ell$ different from $p$. We define “nilpotent lifts” of irreducible generic $k$-representations of $GL_{n}(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$-representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$. This gives a characterization of the mod-$\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

We compute cones of effective cycles on some blowups of projective spaces in general sets of lines.

In this paper, I would like to point out some characteristic features of formal systems having just one primitive notion. Most remarkable systems of this kind may be Zermelo’s and Fraenkel’s set-theories, both having just one primitive notion ∈.

Let L be a fixed language and K be a set of structures related to L. A sentence in L is said to be K-valid if and only if for every structure in K,

Let p be an odd prime number. By using Iwasawa theory, we shall construct cyclotomic fields whose maximal real subfields have class group with arbitrarily large p-rank and conductor with only four prime factors.

Let D be a plane domain and Γ be a component of the boundary of D consisting of a single point. According to Sario [5] we shall call Γ a weak boundary component of D if its image under any conformal mapping of D consists of a single point. A weak boundary component has been introduced by Grötzsch [2], who called it “vollkommen punktförmig”. If Γ is not weak we shall say that it is unstable (Sario [5]). We know that the weakness depends merely on the configuration of D in a neighborhood of Γ (see [4], p. 274).

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.